Speaking for Republicans in a Saturday radio address, Sen. Roy Blunt of Missouri sought to put the burden of a deal on Obama and Reid.
"We still can avoid going over the fiscal cliff if the president and the Democrat-controlled Senate step forward this week and work with Republicans to solve this problem and solve it now," he said.
Whatever manages to pass in the Senate, with its Democratic majority, would then face a second test in the Republican-controlled House.
Boehner, a Republican speaker who has struggled recently with antitax rebels inside his own party, said through an aide that he would await the results of the talks between the Senate and White House. A House vote could come as late as Wednesday, the final full day before a new Congress takes office.
Officials said there was a general understanding that any agreement would block scheduled income tax increases for middle-class earners while letting rates rise at upper-income levels.
Obama was sticking to his campaign call for increases above $250,000 in annual income, even though in recent negotiations he said he could accept $400,000.
The two sides also confronted a divide over estate taxes. Obama favors a higher tax than is currently in effect, but one senior Republican, Sen. Jon Kyl of Arizona, said he's "totally dead set" against it. Speaking of fellow GOP lawmakers, he said they harbor more opposition to an increase in the estate tax than to letting taxes on income and investments rise at upper levels.
But the estate tax was more likely to be used as a possible bargaining chip that Democrats could give away in exchange for higher rates for top earners and other Obama priorities.
Obama and Democrats want to prevent the expiration of unemployment benefits for about 2 million long-term jobless men and women, and there is widespread sentiment in both parties to shelter doctors from a 27% cut in Medicare fees.