In the House, where Mr. Biden can currently afford to lose only eight votes, a Democratic congressman from New York warned that he would not support the president’s plan unless it eliminated a rule that prevents taxpayers from deducting more than $10,000 in local and state taxes from their federal income taxes. He is one of a handful of House Democrats who are calling on the president to repeal the provision.

And in the Senate, where most major legislation requires 60 votes to advance, Senator Chuck Schumer of New York, the majority leader, was exploring an unusual maneuver that could allow Democrats to once again use reconciliation — the same fast-track budget process they used for the stimulus plan — to steer his spending plans through Congress in the next few months even if Republicans were unanimously opposed.

While an aide to Mr. Schumer said a final decision had not been made to pursue such a strategy, the prospect, discussed on the condition of anonymity, underscored the lengths to which Democrats were willing to go to push through Mr. Biden’s agenda.