In Massachusetts, Gov. Charlie Baker didn’t vote for Mr. Trump and has been a frequent critic of the former president. He hasn’t announced whether he will run for re-election. But Mr. Baker left allies with the impression that he was more inclined to seek a third term after Mr. Trump endorsed a potential primary rival last month in former state Rep. Geoff Diehl, who was co-chairman of Mr. Trump’s 2020 campaign in the state and lost a U.S. Senate bid in 2020 to Democrat Elizabeth Warren.

Jim Conroy, Mr. Baker’s political adviser and former campaign manager, said the governor’s decision-making process would be informed by “what is best for Massachusetts and for his family—not by the endorsements of politicians.”

In Ohio, Gov. Mike DeWine —who supported Covid-19 restrictions longer than Mr. Trump and has said the former president bears some responsibility for the Jan. 6 riot in the U.S. Capitol—has drawn a challenge from Rep. Jim Renacci, a close Trump ally.

In Arizona, Mr. Trump has endorsed Kari Lake, a former TV broadcaster who has falsely said the 2020 election was “stolen,” to replace Gov. Doug Ducey, who opposed Mr. Trump’s bid to overturn the election.