"I think the president is eager to go over the cliff for political purposes," Sen. John Barrasso, R-Wyo., said on "Fox News Sunday."
"I think he sees a political victory at the bottom of the cliff," said Barrasso, the third-highest ranking GOP senator. "He gets all these additional tax revenue for new programs. He gets to cut the military, which Democrats have been calling for, for years, and he gets to blame Republicans for it."
Barrasso said he believed no deal would be reached and the United States would go over the so-called cliff, which is more than $500 billion in spending cuts and tax increases set to go into effect in eight days.
Economists have said this could bring the United States back into a recession.
The White House had no immediate comment on Barrasso's remarks.
Obama, in Hawaii with his family for Christmas, earlier called on Congress to "cool off" over the holiday break, amid rising rhetoric on both sides.
He and House Speaker John Boehner, R-Ohio, both said they intended to be back in Washington after Christmas to work on at least a stripped-down deal to beat the cliff's Jan. 1 deadline.
Obama said he and congressional leaders could work on long-term deficit reduction afterward.
The president has proposed at minimum extending current tax rates on middle-income taxpayers and extending an expiring unemployment-benefits program.
The House went home for Christmas break Thursday after Republicans abandoned Boehner's backup plan, known as Plan B, which would have extended current tax rates for income below $1 million.
Other lawmakers said Sunday they thought it wasn't too late to avert the cliff, although they acknowledged time had all but run out for a comprehensive deal.
"I think there's unfortunately only going to be a small deal," Sen. Mark Warner, D-Va., said on the CBS News program "Face the Nation."
Added departing Sen. Kay Bailey Hutchison, R-Texas: "It is going to be a patch, because in four days we can't solve everything. But I think we need to stop this fiscal cliff at a reasonable salary level and then start working on the spending cuts."