President Trump quietly met Sunday in the White House with Georgia Gov. Brian Kemp and the governor’s preferred pick to fill the state’s Senate seat, Atlanta businesswoman Kelly Loeffler.

But the private huddle turned tense and ended quickly, according to people familiar with the meeting. Mr. Trump has preferred Rep. Doug Collins, a Republican who has vocally defended the president during the impeachment process, and he told Mr. Kemp that he would be taking a risk by appointing the politically untested Ms. Loeffler. At one...