President Barack Obama already has scheduled a White House event on Tuesday with some whose benefits expired at the end of December.
"Instead of punishing families who can least afford it, Republicans should make it their New Year's resolution to do the right thing and restore this vital economic security for their constituents right now," Obama said Saturday in his weekly radio and Internet address.
Republicans hinted they might go along with extending benefits if they win spending cuts from Reid elsewhere to pay for them.
"If the senator comes up with any kind of a reasonable idea to offset the $26 billion, I think that he might find some people that are willing to talk to him," said Rep. Matt Salmon, R-Ariz.
Schumer, one of his party's leaders, said Democrats would prefer to pass the proposal as is a without a way to pay for it, as has been the case for previous extensions. But he told reporters Sunday he would listen to GOP suggestions.
During a separate interview, Reid predicted widespread inaction would be the norm "unless the Republicans in Congress decide they should do something for the American people, I'm sorry to say."
Such rancor ruled in the first session of the 113th Congress with few bills passed and sent to the president. The combination of divided government and the upcoming elections stand as an obstacle to major legislation in the second session, counting down to November when all 435 House seats and 35 Senate seats will be on the ballot.
Still, Congress must deal with some significant unfinished business before delving deep into political votes and extended breaks for campaigning.
The Senate was to vote Monday on Obama's nomination of Janet Yellen to become the head of the Federal Reserve. If confirmed, Yellen would become the first woman to fill the powerful post, replacing Ben Bernanke.