Democrats appear likely to include those proposals in their platform. “Senator Sanders, working with other progressives, has transformed the debate, so that expanding Social Security is today a central and consensus tenet of the Democratic Party,” Nancy J. Altman, the president of Social Security Works, an advocacy group, said at a hearing of the platform-drafting committee on June 9.

Representative Kevin Brady, Republican of Texas and chairman of the Ways and Means Committee, said the Democrats’ stance was not a surprise. “Whenever there’s a problem with a federal program,” Mr. Brady said, “the president’s answer is to raise taxes and throw more money at it.”

Mr. Sanders has long pushed for the expansion of Social Security and made that a major theme of his bid for the Democratic presidential nomination. The party’s new consensus gave Mr. Sanders reason to crow.

“A few years ago we were told that the debate on Social Security was not whether we were going to cut it, but by how much,” said Warren Gunnels, the policy director for the Sanders campaign. “Many of us stood up and fought back.”

Now, he said, after Mr. Obama’s remarks this month in Indiana, “we are all unified as Democrats, and the issue is not whether we’re going to expand Social Security, but how much we’re going to expand it.”

Sixty million people receive Social Security benefits totaling more than $74 billion each month. So even small changes can have big implications.

At one point, just before the New Hampshire primary in February, Mr. Sanders challenged Mrs. Clinton’s record on Social Security and suggested that she might support increasing the age at which people become eligible for full retirement benefits. (That age, originally set at 65, increases gradually and is already scheduled to reach 67 for people born after 1959.)