Tennant, who is currently working on a production of Hamlet, has been followed to Stratford-upon-Avon by the mystery woman, claims the Daily Star.
The stalker has reportedly bought tickets for the show every weekend until November and has started sending Tennant sci-fi versions of the play, featuring him and her in the lead roles.
A source said: "Obviously, David gets crowds of fans. But he’s never experienced anything as unsettling as this.
"She has been hassling cast members of the production. And she’s told them that she’s left her husband for David. She regularly leaves pencil drawings for him of them being friendly in the Tardis.
"And she writes bizarre sci-fi versions of Hamlet, featuring herself as Ophelia and David as a futuristic Hamlet.
"She insists that both the Doctor and Hamlet are aliens in a hostile universe, who are terrified of being alone. But she will save him by being his Rose or Ophelia.
"Sadly, the only thing she and Ophelia have in common is their madness. And David is afraid, very afraid."