Of all the money raised by the bill, about one-fourth would be used to increase benefits, and the rest would cover projected deficits in the Social Security trust over the next 75 years.

Democrats, weary of political fights over the Affordable Care Act and divided over a “Medicare for all” single-payer health care plan, relished the opportunity to rally around Social Security. They channeled the spirit of Franklin D. Roosevelt, introducing their bill on the 137th anniversary of his birth, while standing in front of a life-size cardboard cutout of him.

The House speaker, Nancy Pelosi, “supports Democrats’ efforts to strengthen Social Security and bolster seniors’ retirement security,” said Henry Connelly, a spokesman for Ms. Pelosi. She strenuously and successfully opposed efforts by Mr. Bush to divert some payroll taxes into the private retirement accounts favored by many Republicans.

Nancy J. Altman, the president of Social Security Works, an advocacy group, said: “Expanding Social Security, with no cuts, is wise policy and winning politics. People are worried about a retirement income crisis, and this is a solution.”

Senators Richard Blumenthal of Connecticut and Chris Van Hollen of Maryland, both Democrats, are leading a push for the bill in the Senate. A strong vote for the bill in the House, combined with political pressure in a presidential election year, could create momentum for the bill in the Senate in 2020. In any event, it provides a template for future efforts to shore up Social Security.

Republicans say that entitlement programs, not tax cuts, are the main reason for growing deficits. They have in recent years suggested unpopular changes in Social Security, including an increase in the retirement age and a new formula for computing cost-of-living adjustments that could reduce future benefits.

As a presidential candidate, Mr. Trump said, “I’m not going to cut Social Security like every other Republican.” Mr. Larson praised Mr. Trump for that stand and said, “It’s no surprise that he carried the senior vote.”