Republicans’ latest proposal includes a nominal drop in total discretionary spending next year. But that cut is not evenly distributed; in their plan, military spending would continue to rise, while other programs would face deeper cuts.

Mr. Biden’s offer would set spending caps for two years. Republicans would set them for six years.

Republicans have also proposed several efforts to save money that White House officials have objected to. They include new work requirements for recipients of Medicaid and the Temporary Assistance for Needy Families program. They would also make it harder for states to seek waivers for work requirements for certain recipients of federal food assistance who live in areas of sustained high unemployment — a proposal that was not in the Republican debt-limit bill that passed the House.

Republicans are also continuing to seek a reduction in enforcement funding for the I.R.S., a move that the Congressional Budget Office estimates would increase the budget deficit by decreasing future federal tax receipts. And they have sought to include some provisions from a stringent immigration bill that recently passed the House, according to a person familiar with the proposal.

“We are all concerned about deficits and fiscal responsibility, but deficits can be addressed both through changes in spending and through changes in revenue,” Ms. Yellen said, adding that she was “greatly concerned” about Republican proposals to cut funding for the I.R.S.

Mr. Biden insisted on Sunday that he was willing to cut spending. He also suggested that some Republicans were trying to crash the economy by not raising the borrowing limit, in order to hurt Mr. Biden’s hopes of winning re-election.