WASHINGTON—President Biden on Wednesday extended through Aug. 31 the pause on payments of federal student loans, setting up a politically fraught decision about whether to resume payments less than three months before the midterm elections and prompting renewed calls from Democrats to cancel student debt.

Mr. Biden has so far resisted calls by progressive members of his own party to forgive student debt on a large scale. The issue has animated young liberal voters, who have embraced a proposal by Sen. Elizabeth Warren (D., Mass.) to cancel up to $50,000 in student debt per borrower.