Gaga, 25, real name Stefani Germanotta, is reportedly worried that her level of fame is so high that something similar could happen to her.
A source told British newspaper The Sun: "Gaga has always been absolutely fixated on Diana and sees herself as a 21st-century incarnation of her.
"She knows she is almost as recognisable as Diana and is hounded by photographers and fans on the same level.
"Gaga has had dreams about dying the same way as her and has genuine fears that her fame could literally kill her, either in a chase or at the hands of crazies."