Republicans say they have already given ground on more tax revenue, and any future deficit reduction must come from spending cuts.
In the Senate, a group of Republicans on the Armed Services Committee proposed delaying the spending cuts until the 2013 fiscal year ends on Sept. 30.
Their plan would replace the $85 billion in cuts with savings achieved by shrinking the federal workforce by attrition over a multi-year period, an approach unacceptable to Democrats.
As government workers left their jobs, they would not be replaced. The plan also would extend a congressional pay freeze that was put in place as part of the Jan. 1 fiscal cliff deal.
Those lawmakers said their plan would protect vital defense programs and military readiness.
Sen. Lindsey Graham of South Carolina issued a challenge to fellow Republicans willing to accept the defense cuts in the sequester: "If you feel comfortable with cutting the government this way, then you have lost your way as much as the president."
"I am sure Iran is very supportive of sequestration. I am sure al-Qaida training camps all over the world must be pleased with the fact that sequestration will gut the CIA and the intelligence platforms that follow them around," Graham said.
Lawmakers whose districts have heavy concentrations of military installations and defense manufacturing are growing nervous about the looming cuts.
Rep. Rob Bishop, a Republican member of the House Armed Services Committee, said he would support delaying the cuts without any offsetting savings.
"I'll take any kind of postponement I can get," said Bishop, who is worried about the defense workers in his Utah district. "I feel for my constituents who have had their salaries frozen for years."