It comes after Piers recently interviewed Trump in an explosive chat to launch his Uncensored show on TalkTV earlier this year.
Trump recently hit the headlines again as the FBI searched his Mar-a-Lago estate in Florida to investigate boxes containing classified information.
Both Sugar and Morgan have been embroiled in a public feud that often sees them engage in a back and forth on social media.
Last week, Alan brutally mocked Piers over his “toothpick leg” after the former Good Morning Britain star uploaded a picture of himself on Twitter.
The Apprentice star turned his attention to one of broadcaster Piers’ legs sticking out of his shorts and seeming unusually thinner than the other one.