Many leading Republicans have already concluded that Mr. Trump is sure to lose, and that the party should turn its attention entirely to buttressing its most endangered senators and limiting Democratic gains in the House. But there is a divide over how soon the Republican National Committee should begin shifting money away from the presidential race and to the fight to retain Congress.

“We’re rapidly approaching the time where the R.N.C. will have to think long and hard about investing its resources in Senate seats rather than continuing to help a presidential campaign that’s going nowhere,” said Josh Holmes, a Republican strategist deeply involved in Senate races.

But at a private meeting of the Republican Governors Association this week in Aspen, Colo., Haley Barbour, an influential former Mississippi governor and Republican chairman, urged the governors to rally behind Mr. Trump to improve the party’s prospects and said the party should continue to back him, at least until October, according to a Republican who was there.

Democrats would need to pick up four seats to gain control of the Senate if Mrs. Clinton wins the presidency, and her vice president becomes the chamber’s tiebreaker vote. Officials in both parties believe that the two Republican seats most likely to change hands are in Wisconsin and Illinois. And with the well-known former Senator Evan Bayh of Indiana returning home to try to reclaim his Republican-held seat, Democrats enjoy an initial advantage there, too.

So with only one Democratic-controlled seat being aggressively contested — that of the Senate minority leader, Harry Reid of Nevada, who is retiring — it is critical for Republicans to bolster their defenses in Pennsylvania, New Hampshire and North Carolina, or they are sure to lose the Senate.