While Mr. Sanders is widely credited with bringing the idea of free college to national attention during his 2016 presidential run, Ms. Warren pushed the idea further in April with an additional call for the elimination of student debt. She has siphoned away parts of Mr. Sanders’s base of voters with a stream of progressive plans. Julián Castro, a former housing secretary and current presidential candidate, has also unveiled a plan to transform the student loan repayment system.

Ms. Warren’s $1 trillion plan would eliminate some or all student loan debt for about 95 percent of borrowers — about 42 million people, according to her campaign — canceling up to $50,000 for every person with a household income of less than $100,000. More than 75 percent of borrowers would see all of their student debt eliminated. She has said that she plans to introduce accompanying legislation with Representative James E. Clyburn of South Carolina, the No. 3 Democrat in the House.

But Mr. Sanders’s legislation makes no distinctions for income levels.

“Rather than making exceptions, let’s end this crisis entirely once and for all,” said Ms. Omar, who made the cancellation of student loans a prominent issue during her 2018 campaign. “The American people bailed out Wall Street. It’s time for Wall Street to bail out the American people.”

Mr. Sanders told reporters he did not support an income cutoff because “I believe in universality.”

“If Donald Trump wants to send his kids to public schools,” he added, “he has a right to do that.”

Some more moderate analysts were measured about the plan, asking for more detail on its funding, state involvement and a more direct antidote to the rising costs of secondary education.

The plan “doesn’t match the complexity of the cost education problem,” said Joni E. Finney, a professor at the University of Pennsylvania and the director of the school’s Institute for Research on Higher Education. “This is providing it all on the back end, and doesn’t deal with the angst students have on the front end.”

Others in the education field, like Randi Weingarten, the head of the American Federation of Teachers, backed it.