WASHINGTON—When Donald Trump and Ted Cruz took their last turns onstage during the 2016 primary campaign, the senator was booed for refusing to offer his endorsement of the would-be president during the Republican National Convention.

On Monday, President Trump will travel to Houston to offer his endorsement for Mr. Cruz’s re-election bid this November. The president will tell Texas voters that he needs Mr. Cruz—and Republicans like him—to help toughen immigration laws, repeal the Affordable Care Act and fill the courts with...