Although Republican infighting in the House is dominating headlines, the radically divergent trajectory being followed by the GOP in the Senate suggests that the battle within the party has barely begun.

Whoever rises to lead the House majority as speaker will — if that person is a Republican — be beholden to the party’s right flank, which has already demonstrated an appetite for obstructionism and chaos and an unwillingness to compromise.

In the Senate, however, the GOP leader, Senator McConnell, has signaled that his patience for the sort of grandstanding going on at the other side of the Capitol is wearing thin.

The best look into his thinking comes via his analysis of why Republicans fared so poorly in the 2022 midterms, when they theoretically should have ridden a red wave to undisputed victory.

“We lost support that we needed among independents and moderate Republicans, primarily related to the view they had of us as a party — largely made by the former president — that we were sort of nasty and tended toward chaos,” Mr. McConnell told NBC News.

This statement is only the latest signal that he doesn’t see a path forward for the GOP with Mr. Trump or his style of politics, especially the “chaos,” leading the charge.

The question of Mr. Trump’s future is not the only area in which Mr. McConnell has broken with the right wing of his party. The bigger issue will be in the Senate leader’s willingness to cut deals, something hardliners in the House have signaled they will not do.

Mr. McConnell, for example, voted for the bipartisan infrastructure bill, as well as the most recent omnibus bill that passed in the last Congress, while the House minority leader at the time, Kevin McCarthy, whipped votes against the measures.

Regardless of whether Mr. McCarthy becomes speaker or someone else ends up in the leadership position, the right wing of the GOP will be in the driver’s seat in the House because of the concessions Mr. McCarthy has made so far in his effort to get elected as House leader.

As of Thursday, Mr. McCarthy has essentially caved on every demand made of him, including ones that will likely make governance in the House nearly impossible given that any five Republicans now will be able to block any legislation.

Even Wednesday’s vote to adjourn the House after three failed speaker votes proved to be nearly too contentious to pass, squeaking by on only two votes.

Mr. McCarthy’s latest round of concessions paint a picture of a House where this sort of chaos will be the norm rather than the exception.

Mr. McCarthy also has reportedly acquiesced to the idea of a “motion to vacate,” meaning any single member of the House can trigger a vote of no confidence in the leader’s standing at any time.

He has also agreed to give two seats on the House Rules Committee to members of the House Freedom Caucus, the same group that is currently opposing him, and the Freedom Caucus members are demanding that they get to choose the two members.

If this comes to fruition, then the lawmakers that a conservative congressman, Dan Crenshaw, called “clowns” after one House session would be greatly empowered in how the new Congress operates.

Unless a deal is cut with Democrats, who currently show no signs of wanting to bail out the GOP, any speaker candidate will have to match Mr. McCarthy’s offer to the Freedom Caucus.

Although Mr. McConnell has voiced his support for Mr. McCarthy in the speaker fight, the terms that Mr. McCarthy has offered will all but guarantee chaos in the House next year.

Beyond the procedural changes, Mr. McConnell has taken a stance on the debt ceiling that diverges from the one preferred by Republicans in the House. Congressman Ralph Norman said Wednesday the ceiling is a key concern to the caucus, telling reporters Wednesday, “Us 20 want changes, and we’re going to stay here until we get it.”

“Is he willing to shut the government down rather than raise the debt ceiling?” Mr. Norman said. “That’s a non-negotiable item.”

Mr. McConnell, on the other hand, has historically cut deals over the debt limit, and in December made moves — unsuccessful, as it turned out — to try to raise the debt limit during the lame duck session.

All of these conflicts on the horizon for the House and Senate GOP leadership come as the 2024 presidential election draws nearer, and with it a looming fight over the party’s relationship to Mr. Trump, who remains the preferred candidate among many in the Republican base.