ATLANTA—Georgia Republican Gov. Brian Kemp is resisting pressure from President Trump on who to appoint to an interim U.S. Senate post, according to people familiar with the selection process.

In recent days, the president has spoken to Mr. Kemp at least twice—once face-to-face in Atlanta and once on the phone—urging him to pick Rep. Doug Collins (R., Ga.), a vocal supporter of the president in Congress, these people said. Mr. Collins, a white conservative from north Georgia, has pushed for months to get the seat that Sen....