The X Factor presenter was believed to have raised the alarm after picking up the threat on his voicemail left by an unknown caller.
A City of London Police spokesman confirmed an investigation had been launched into an “offensive message”.
“We are investigating claims of an offensive message being left on a mobile telephone. It follows a complaint from a member of the public,” the spokesman said.
Security for The X Factor judges was increased in 2008 after a tracking device was put into Simon Cowell’s car.