Word of the request got back to Mr. Priebus and his associates, who bridled at the nature of Mr. Stivers’s appeal in part because they feel that the administration has been generous to House Republicans, the officials said. Mr. Trump headlined a March fund-raising dinner for the National Republican Congressional Committee that brought in $30 million. Late last month, the president was the keynote speaker at a fund-raiser in Atlanta for Karen Handel, the Republican nominee in the race to replace Mr. Price.

Other Republicans, however, say that Mr. Stivers did not demand anything of the R.N.C. The R.N.C. is still expected to transfer money to the House campaign committee for the special elections.

The race to replace Mr. Zinke in Montana will be decided on May 25, and both parties believe the Republican nominee there, Greg Gianforte, enjoys an advantage.

The runoff between Ms. Handel and the Democrat Jon Ossoff for Mr. Price’s old seat, which will take place on June 20, is more unpredictable. Mr. Ossoff received just over 48 percent of the vote in the first balloting last month, energizing Democrats and anti-Trump Republicans. The two House campaign committees, along with a handful of outside groups, have poured money into the race, making it the most expensive in the House’s history.

Tensions between the White House and the party campaign committees are hardly novel, and they often flare when a president is saddled with flagging approval ratings. But the strains are mounting quickly in this case in part because Mr. Trump, an unconventional political leader, lacks the sort of relationships with Republican officeholders that most presidents enjoy upon entering the Oval Office.

Further, in this White House, there is no go-to political point person — a role that Karl Rove and David Axelrod filled in previous administrations — and no high-ranking official overseeing the president’s role in the party and campaigns. Mr. Priebus has tried to play that part, but as the top aide to an impulsive and improvisational president, he has other responsibilities that consume his time.