WASHINGTON—The White House said Tuesday that President Donald Trump believes he has the authority to fire special counsel Robert Mueller, as lawmakers from both parties warned against doing so one day after the FBI raided properties tied to the president’s longtime lawyer.

Senate Majority Leader Mitch McConnell (R., Ky.) said that Mr. Mueller “should be allowed to finish his job,” though he rebuffed calls for legislation to protect the special counsel. Sen. Chuck Grassley (R., Iowa), chairman of the Judiciary Committee, said...