As the dust clears after Congressman Kevin McCarthy’s bruising fight to become the speaker of the House, the pressing question is whether the deal he cut to get that job will result in cuts to defense spending.

The new bargain is not mentioned in a new slate of rules that would end the practice of voting on annual budget bills through an omnibus process that lashes spending and other legislation together for an up-or-down vote, usually at the end of the fiscal year.

Rather, House sources tell me, the deal on defense spending was part of a handshake agreement between the new speaker and the Republican holdouts who ended up voting him in as speaker in the 15th round over the weekend.

In theory, that agreement would limit House appropriations for discretionary spending to fiscal year 2022 levels. That would cap defense spending at $782 billion, still more than China and Russia, but significantly less than the $857 billion slotted for 2023.

Stirring the pot, Congressman Jim Jordan, a Republican of Ohio, told Fox News Sunday that defense spending cuts would be on the table for the new Congress. “We got a $32 trillion debt. Everything has to be on the table,” he said.

The former general counsel to the House Armed Services Committee, Roger Zakheim, Washington director of the Ronald Reagan Institute, told me that the overall effect of capping defense spending at 2022 levels would be a 13 percent cut in the budget from 2023 levels, when factoring in 5 percent inflation.

“When you get in that scenario, you force military planners to take away money for modernization to focus on sustaining the force you have now,” Mr. Zakheim said.

Congressman Mike Gallagher, a Republican of Wisconsin who serves on both the House Intelligence and Armed Services Committees, came out in support of the new House rules package. He also said that the new House rules do not guarantee cuts to defense spending.

“Anyone suggesting this package cuts defense spending is ignoring the math,” Mr. Gallagher said. “There are not 218 members that support defense cuts and any budget resolution that tries to do so will fail.”

Congressman Chip Roy, a Republican of Texas who was one of the leaders of the Republican holdouts in the fight over the speaker, said on Twitter Sunday that the deal with the new speaker did not mandate any cuts for defense spending.

Instead, he said the deal meant focusing on cuts to non-defense spending in the $1.7 trillion annual discretionary budget for the federal government. This is the part of the federal budget that does not include entitlement programs like Medicare and Social Security.

Part of the concern for national security hawks in Congress is that the agreement that Mr. McCarthy made to get the speakership gives the option for members of his own caucus to call for leadership elections whenever they choose. That looks like a Damocles sword hanging over the new speaker.

Even a few disgruntled Republican members could bring House business to a standstill. While a large majority of Republicans support a strong defense, there are some who do not.

Last week as Mr. McCarthy was twisting in the wind after losing votes to secure his post, Congressman Matt Gaetz, a Republican of Florida, tweeted that the biggest loser in the disarray was Ukraine’s president, Volodymyr Zelensky, and the biggest winner was the American taxpayer.