The message, which was intercepted by Gaga's management team, read: "You come to my dreams. I want to die and I want to die together with you.
"I am not (John Lennon's assassin) Mark Chapman. It is not only you who will die. I will shoot a bullet through my head too."
The 26 year old, who lives in New York, has been ordered to keep her distance from Gaga and stay at least 500 yards (457 metres) away from her at all times.
Mark Chapman is behind bars after shooting the Beatles legend dead outside his New York apartment in 1980.