Some believe the most effective plan would be a small-scale patch since time is running out before the year-end deadline.
Even those who believe a deal is possible are not expecting a lot.
Sen. Kay Bailey Hutchison, R-Texas, said she thinks a patch is the best bet, saying that there's a slim chance of solving the issue in just four days.
Talks to avoid the cliff are set to continue on Thursday. President Barack Obama and Congress are on a short holiday break. Congress is expected to be back at work Thursday and the president will be back in the White House after a few days in Hawaii.
Some Republican lawmakers blame Mr. Obama for the stalemate saying he wants the country to go over the cliff.
Last Thursday, a vote on House Speaker John Boehner's "Plan B" was scrapped. It sought higher taxes on people making over $1 million a year.
If Congress fails to reach any agreement, income tax rates will go up on just about everyone on Jan. 1.
Mr. Obama has scaled back his ambitions for a sweeping budget bargain. On Friday, he called for a limited measure that extends George W. Bush-era tax cuts for most people and staves off federal spending cuts. The president also urged Congress to extend jobless benefits for the long-term unemployed that would otherwise be cut off for 2 million people at the end of the year.
Last week, news emerged that Mr. Obama and Boehner had significantly narrowed their differences. Both were offering a tax cuts for most Americans, an increase for a relative few and cuts of roughly $1 trillion in spending over a year.
The Associated Press contributed to this report.