HOLLYWOOD, Fla.—Donald Trump and his top aides have been trying privately to offer an olive branch to a GOP establishment that he has vilified. Many party insiders are wary of grasping it.

Reince Priebus, chairman of the Republican National Committee, tried to bridge that divide Friday with a double-edged message: The stop-Trump forces should lay down their weapons, and Mr. Trump should stop attacking the party.

“Politics...