Ebullient Republicans returned to Capitol Hill on Tuesday after elections that put them on the brink of taking control of both chambers of Congress to face critical questions about how they will wield their power — and how tight a grip President-elect Donald J. Trump will have on their new majority.

G.O.P. senators were set to make a monumental choice on Wednesday, when, for the first time since 2007, they plan to elect a party leader not named Mitch McConnell. Three men have been quietly jockeying for months to replace Mr. McConnell, the longest-serving Senate party leader in history, but some of Mr. Trump’s allies have pressed him to block the man considered to be the front-runner: Senator John Thune of South Dakota.

At the same time, Mr. Trump’s push to stack his administration with loyal members of Congress was colliding with a tough political reality for Republicans: They are running out of the bodies they need to preserve the narrow House majority they expect to hold.

With the party on track to win the House by a precariously small margin and Mr. Trump tapping two House members to serve in top national security posts, Republican leaders warned on Tuesday that they could not spare any more.

“I don’t expect that we will have more members leaving, but I’ll leave that up to him,” Mr. Johnson said, referring to Mr. Trump.

Control of the House of Representatives is still officially up for grabs, with several competitive races not yet called, but Republicans appear positioned to keep the majority by a similarly tiny edge that has made it so difficult for them to govern over the past two years. As of Tuesday afternoon, they had won control of 214 seats, and Democrats had 205. A party needs 218 seats for a majority, and most strategists expect that the G.O.P. will not end up with many more than that.