The leader of the American Federation of Teachers, Randi Weingarten, sees some of the proposals, in particular the pension issue, as an attack on teachers. She also has influence over more than $1 trillion in public-teacher pension plans, many of which traditionally invest in hedge funds.

It is a recipe for a battle for the ages.

Ms. Weingarten started by targeting hedge-fund managers she deemed a threat to teachers and urged unions to yank money from their funds. Then she moved to Wall Street as a whole.

Her union federation is funding a lobbying campaign to eliminate the “carried-interest” tax rate on investment income earned by many money managers. It is trying to defeat legislation that would increase the charitable deduction in New York state for donations to private schools. And it has filed a class-action lawsuit accusing 25 Wall Street firms of violating antitrust law and manipulating Treasury bond prices.

‘Given your strong support…for an organization which is leading the attack on defined benefit (DB) pension funds around the country, I was surprised to learn of your interest in working with public pension plan investors.’
—Randi Weingarten, in March 15, 2013, letter to hedge-fund manager Daniel Loeb

Some pension funds have withdrawn money from hedge-fund managers criticized by the teachers union. And some hedge-fund managers stopped making donations to advocacy groups targeted by Ms. Weingarten.

Hedge funds, reluctant to buckle to the pressure, say Ms. Weingarten is doing a disservice to the teachers she represents, because funds should aim solely to earn the highest possible return on their assets. The personal beliefs or donations of hedge-fund managers, they argue, shouldn’t be a factor in that decision. At least one manager, Mr. Loeb of Third Point LLC, has increased his donations to a charter-school group, citing Ms. Weingarten.

Sander Read, chief executive officer of Lyons Wealth Management, which hasn’t been targeted, likened what Ms. Weingarten is doing to “hiring a dentist because of their political beliefs. You may see eye to eye on politics, but you may not have great, straight teeth.” None of the hedge funds targeted by the teachers unions would discuss the matter publicly, a sign of how sensitive the battle has become.

Ms. Weingarten said in an interview: “Why would you put your money with someone who wants to destroy you?”

The battles are rooted in a political fight over how to improve public education. Republicans have long sought major changes, such as creating new competition for public schools, including charter schools. Democrats largely have supported solutions backed by the unions, particularly increased spending for existing schools.

About a decade ago, some liberals joined conservatives in pushing to expand charter schools. Those efforts received financial support from hedge-fund managers including Mr. Loeb, Mr. Singer of Elliott Management Corp. and Paul Tudor Jones of Tudor Investment Corp., who together kicked in millions of dollars.

Some of those involved in the effort cast public-school teachers and their unions as obstacles to improving education. The reputation of the teachers union took a beating.

When Ms. Weingarten was elected president of the American Federation of Teachers in 2008, she aimed to restore public trust in public-school teachers and their unions.

As she rose in the union, she got close to Bill and Hillary Clinton. Last summer, the federation became the first union group to endorse Mrs. Clinton’s presidential campaign. Ms. Weingarten sits on the board of the super PAC supporting her candidacy, and the American Federation of Teachers has donated $1.6 million to the Bill, Hillary and Chelsea Clinton Foundation.

Ms. Weingarten’s federation represents about two dozen teachers unions whose retirement funds have a total of $630 billion in assets, a big chunk of the more than $1 trillion controlled by all teachers unions. The federation doesn’t control where that money is invested; the unions themselves do. But Ms. Weingarten can make recommendations.

She instructed investment advisers at the federation’s Washington headquarters to sift through financial reports and examine the personal charitable donations of hedge-fund managers. She says she focuses on groups that want to end defined-benefit pensions. Many of the same entities also back charter schools and overhauling public schools.

In early 2013, the union federation published a list of roughly three-dozen Wall Street asset managers it says donated to organizations that support causes opposed by the union. It wanted union pension funds to use the list to decide where to invest their money.

The Manhattan Institute for Policy Research, a think tank that supports increasing school choice and replacing defined-benefit pension plans with 401(k)-type plans for future government employees, is one of the groups to which donations were viewed unfavorably.

Lawrence Mone, its president, says the tactics amount to intimidation. “I don’t think that it’s beneficial to the functioning of a democratic society,” he says.

After KKR & Co. President Henry Kravitz made the list in 2013, Ms. Weingarten got a call from Ken Mehlman, an executive at the private-equity firm and former chairman of the Republican National Committee.
Mr. Mehlman said KKR had a record of supporting public pension plans, according to Ms. Weingarten.

Ms. Weingarten agreed, removed Mr. Kravitz’s name from the list and invited Mr. Mehlman to talk about the firm’s commitment to public pensions at a meeting in Washington with 30 pension-fund trustees representing 20 plans that control $630 billion in teachers’ retirement money.

When Cliff Asness of hedge fund AQR Capital Management LLC found out Mr. Kravitz had gotten off the list, he called Mr. Mehlman, a friend. Mr. Asness also hired a friend of Ms. Weingarten’s: Donna Brazile, a vice chairwoman of the Democratic National Committee who has been a paid consultant to the American Federation of Teachers.

Ms. Brazile arranged a lunch meeting between Mr. Asness and Ms. Weingarten, where they discussed ways to work together. Not long after, Mr. Asness’s firm paid $25,000 to be a founding member of a group that KKR’s Mr. Mehlman was starting with Ms. Weingarten to promote retirement security.

Mr. Asness was removed from the list. A year later, when Ms. Weingarten noticed he continued to serve on the Manhattan Institute board, she considered putting him back on.

In September of last year, when the California State Teachers’ Retirement System, or Calstrs, considered increasing its hedge-fund investments, Ms. Weingarten saw another chance to apply pressure.

Dan Pedrotty, an aide to Ms. Weingarten who runs the hedge-fund effort, spoke to a Calstrs official about Mr. Asness’s continued service on the Manhattan Institute’s board. The Calstrs official then called Mr. Asness.

In December, Mr. Asness said he would step down from the Manhattan Institute board. His spokesman says he already had made the decision at the time of the call, after reassessing time spent on the boards of several nonprofit groups.

“Randi is committed to helping hard working employees achieve the secure retirement they deserve,” Mr. Asness said in a written statement.

Mr. Loeb, founder of the $16-billion Third Point fund, has been more combative. He is a donor to the Manhattan Institute and chairman of the Success Academy, which operates a network of charter schools in New York City.

‘I can appreciate that it may be frustrating for the certain plan sponsors to invest with managers who have different political views or party affiliations, an issue they must come to terms with due to their fiduciary responsibilities.’
—Daniel Loeb, in March 22, 2013, email to union leader Randi Weingarten

In a March 2013 letter to Mr. Loeb, Ms. Weingarten noted his support of a group “leading the attack on defined benefit pension funds” and said she was “surprised to learn of your interest in working with public pension plan investors.” Seeking business from union pension funds while donating to the group, she wrote, “seem to us perhaps inconsistent.”

The two agreed to meet.

Mr. Loeb emailed Ms. Weingarten, noting his fund’s average annual return of 21% over 18 years. “I completely respect the political considerations you may have and understand if other factors dictate how funds are allocated,” he wrote.

A week later, Ms. Weingarten wrote back to reiterate that unions were wary of investing with Mr. Loeb “given the political attack on defined benefit funds.”

In response, Mr. Loeb asserted that it must be “frustrating” for unions to invest with funds that “have different political views or party affiliations.” He added: “At least we can rejoice in knowing that as Americans we share fundamental values that elevate individual opportunity, accountability, freedom, fairness and prosperity.”

The meeting was called off, and Mr. Loeb was added to the list.

At a fundraising dinner that May for his charter-school group, Mr. Loeb stood up and said: “Some of you in this room have come under attack for supporting charter-school education reform and freedom in general.” He called Ms. Weingarten the “leader of the attack” and pledged an additional $1 million in her name.

“Both Randi and I believe America’s children deserve a 21st century education, and I hope the day comes when she embraces the positive change created by public charter schools,” Mr. Loeb said recently in a written statement.

In late 2013, state union officials pressed a Rhode Island pension fund to fire Third Point. The following January, the pension fund did just that, pulling about $75 million from Mr. Loeb’s fund. A spokeswoman for the state treasurer said at the time that Mr. Loeb’s fund was too risky.

Roger Boudreau, a member of the teachers union and an elected adviser of the Rhode Island fund at the time, says the donations played a role. “It’s fair to say that those kinds of donations are going to be looked at very critically,” he says.

Around that time, a giant billboard appeared above Times Square. “Randi Weingarten’s Union Protects Bad Teachers,” it read above a picture of her scowling face.

Ms. Weingarten immediately assumed the hedge-funders were behind the attack. The entity listed as the billboard’s sponsor is the Center for Union Facts, a Washington-based advocacy group. The group declines to disclose who paid for the billboard.

“We all guessed it had to be people like Dan Loeb,” Ms. Weingarten says. Mr. Loeb declined to comment.

The billboard kicked off a campaign against Ms. Weingarten by the Center for Union Facts, including radio and newspaper advertisements. “She’s the head of the snake, so it was appropriate to go after her personally,” says the group’s president, Richard Berman.

The ads directed people to a website that said she oversaw a “crusade to stymie school reforms and protect the jobs of incompetent teachers.” It listed her salary and called her a “member of the elite.”

In September 2014, Mr. Berman sent a 10-page letter to lawmakers, union officials and opinion leaders charging that Ms. Weingarten‘s “ineptitude is a threat against America, against hard-working teachers, and especially against our nation’s children.”

Lorretta Johnson, secretary-treasurer of the American Federation of Teachers, responded in a letter to union leaders that Mr. Berman represented a “front group whose mission is to vilify and destroy unions.”