The deal between the New York securities firm and the American Federation of State, County and Municipal Employees means Goldman will appoint a "lead" director, but shareholders won't get a chance to vote at the firm's annual meeting in May on the proposal to replace Mr. Blankfein with an independent chairman.

The union had claimed stripping Mr. Blankfein of his chairman powers would help Goldman repair its reputation and reduce the potential for conflicts of interest. Goldman shareholders voted down a similar proposal in 2010 by a wide margin.

But with the firm facing outside pressure over its ethics and business practices, Goldman executives worried the outcome might be different this time, according to people familiar with the situation. Some executives gave the proposal a 50-50 probability of winning.

Afscme has submitted similar shareholder proposals to split the chairman and CEO duties at J.P. Morgan Chase & Co., American Express Co., Northern Trust Corp. and six other companies. None of those firms has held talks with Afscme on the proposals, the union said.

Inside Goldman, Afscme's proposal last September sparked months of discussions and contingency planning among directors and executives on the firm's powerful management committee. That group includes Mr. Blankfein, President Gary D. Cohn and executives who run all of Goldman's operations.

As part of those talks, Goldman officials considered making Mr. Cohn the company's chief executive and reducing Mr. Blankfein's role to chairman if the union's proposal passed, according to people familiar with the matter.

Such a shift would have abruptly ended Mr. Blankfein's reign as CEO. The 57-year-old has been Goldman's chairman and chief executive since 2006. He has told colleagues that he would rather die at his desk than give up running the firm.

Still, Mr. Blankfein indicated that he was willing to step down as CEO if necessary, these people said. Mr. Cohn is the No. 2 executive at Goldman, giving him an advantage in the race to succeed Mr. Blankfein. A Goldman spokesman said Messrs. Blankfein and Cohn weren't available for comment.

In a statement Tuesday night, a Goldman spokesman said the firm's "board of directors and senior management have not had any discussions or conducted contingency planning around splitting the roles of Chairman and CEO."

Some Goldman executives, including rivals of Mr. Cohn, view his brusque management style and ties to Mr. Blankfein as a liability for the firm at a sensitive time. In the wake of the financial crisis, Goldman has worked hard to dig out from trouble stemming from the firm's bets against the housing market, its sales of complex mortgage-backed securities that blew up, and adverse publicity about high pay levels.

The latest headache for Messrs. Blankfein and Cohn came earlier this month when a Goldman employee in London resigned in an op-ed article that accused the firm of "ripping their clients off." Goldman is investigating his claims.

Some institutional shareholders, especially union-affiliated pension plans, have prodded companies to separate the posts of chairman and chief executive, contending that such moves increase the independence of boards and make CEOs more accountable to directors.

In 2009, Bank of America Corp. shareholders banned the Charlotte, N.C., bank from having the same person as chairman and CEO. Soon after, Kenneth D. Lewis stepped down as chairman, and he resigned eight months later as chief executive.

In February, Goldman negotiated a compromise with the union that allows Mr. Blankfein to keep both posts and postpones until his resignation or retirement the final decision on who will lead the company, the union said. Goldman agreed to appoint at May's annual meeting an independent "lead" director, bringing the firm in line with guidelines backed by shareholder-advisory firms.

The change already is reflected in Goldman's corporate-governance guidelines, which were amended earlier this month. Goldman and Afscme confirmed the agreement in response to questions from The Wall Street Journal.

"We appreciated the constructive talks we had with Afscme," a Goldman spokesman said.

Goldman's negotiations with union officials were led by John F.W. Rogers, the securities firm's board secretary. Mr. Rogers is known for his behind-the-scenes maneuvering in New York and Washington.

Lisa Lindsley, director of capital strategies for the Afscme Employees Pension Plan, with more than $850 million in assets, said that "events subsequent to our withdrawal agreement demonstrate that there are serious cultural issues at Goldman."

The deal "is a step in the right direction," she said. "But it remains to be seen if it is enough." Afscme held 7,101 shares of Goldman as of September.

Also putting pressure on Goldman: About 15% of outstanding shares are held by index funds, which often vote in line with the recommendations of proxy advisers. In addition, new rules no longer allow brokers to vote the shares of their clients, and brokers had generally backed management.

People familiar with Goldman's contingency-planning talks said the firm has accelerated conversations about how to groom and vet its next generation of leaders.

Write to Liz Rappaport at liz.rappaport@wsj.com