The push to spend more is also intensifying, especially on defense. The warnings from the military’s uniformed and civilian leadership have grown increasingly dire — deep cuts in personnel that would leave the smallest Air Force in history, tens of thousands of soldiers taken from the combat-ready units and dramatically scaled-back naval rapid-response forces. Democrats say vital nondefense programs are no less imperiled.

Senator Jack Reed of Rhode Island, the ranking Democrat on the Senate Armed Services Committee, said naval warships in the Arctic are only as effective as the Coast Guard icebreakers funded by the Department of Homeland Security that clear their path. The military can track drug shipments from South America, he added, but without a properly funded Justice Department, F.B.I. or Coast Guard, the homeland is not secure.

But most conservative Republicans are not budging in the face of the sky-is-falling warnings. The most frightening predictions for sequestration did not pan out when the cuts hit in 2013. A bipartisan budget deal that year offered some relief through 2015, but when that deal expires Sept. 30, most Republicans are not eager to extend it. Even under the sequester, the limit on defense spending would be $523 billion for the 2016 fiscal year — slightly higher than the limit of $521 billion for 2015.

The strict caps “are one of the best things that’s happened to the finances of the country,” said Senator Jeff Sessions of Alabama, a senior Republican on the Budget Committee.

Senator Rand Paul of Kentucky, who is expected to seek the Republican presidential nomination, said he would not support lifting the caps. Lawmakers are scrambling to finesse the problem. Senator Kelly Ayotte, Republican of New Hampshire, and Mr. Enzi are pressing for a place holder in the budget — a “deficit-neutral reserve fund” — that they say would allow Congress to come back in the coming months with legislation to lift the spending caps.

The idea is to pass a budget this month that sticks to the spending caps, but then negotiate a budget law this summer that ends sequestration. The $540 billion in cuts still to come under the Budget Control Act would be replaced by savings from entitlement programs like Medicare and Social Security as well as new revenue from closing some tax loopholes.

“The budget resolution is not going to allow us to change the law,” Ms. Ayotte said.

To some Republicans — and most Democrats — a “deficit-neutral reserve fund” is a grand deception. Mr. Graham, a member of both the Budget and Armed Services Committees, said he would vote the budget out of committee this week, but without assurances that spending would rise, he would not vote for its final passage.