WASHINGTON—President Donald Trump supports efforts to strengthen the federal background-check system for firearms, his press secretary said Monday, adding that he had spoken with a top GOP senator about legislation to do so following last week’s deadly school shooting in Florida.

White House press secretary Sarah Huckabee Sanders said in a statement Monday that Mr. Trump spoke Friday to Senate Majority Whip John Cornyn (R., Texas) about legislation he and Sen. Chris Murphy (D., Conn.) have proposed to strengthen the background...