PALM BEACH, Fla.—President Donald Trump called together several top advisers Saturday, including Attorney General Jeff Sessions and political strategist Steve Bannon, as the White House faced a growing furor over the president’s early morning claims he had been wiretapped by his predecessor.

Mr. Trump, who is spending the weekend at his private Mar-a-Lago club here, added a meeting with Mr. Sessions and Homeland Security Secretary John Kelly to his calendar on Saturday, as political allies and foes of the president continued...