"I now live in fear," the woman said. "I have moved and left my old life behind. I worry every day that I may be a target for this man."
The woman, whose full name is not being revealed, said a man rang her six months before Ms Dando's death, and asked if she was Jill Dando.
He said he wanted to get in contact with her
because he was a fan but the woman told him she had no idea how to contact the TV star.
"I now live in fear," the woman said. "I have moved and left my old life behind. I worry every day that I may be a target for this man."
Police hunting Jill Dando's killer revealed that the call had been made last week.
"The intention of releasing information about the phone call to a J Dando was as an example of people who are known to have displayed obsessive behaviour about Miss Jill Dando who are yet to be identified," a Scotland Yard spokeswoman said.
Professional killing
The woman spoke to the Sunday Mirror newspaper about her experience.
Police are also investigating new theories that an obsessed policeman or soldier may have killed Ms Dando.
"Police are looking for someone with a knowledge and experience or interest in firearms, perhaps a gun club member or connected by his occupation or previous occupation," the Scotland Yard spokeswoman said.
"Ms Dando was shot once to the left side of the head at very close range. We have always said one must keep an open mind about whether the suspect was an obsessive or it was a professional killing."
The latest round of speculation over the murder comes after a renewed appeal for information broadcast on BBC's Crimewatch programme on Wednesday.
Detectives have been given 200 new names of which 20 are said to be "relevant and interesting".
The tip-offs came after the programme received more than 400 calls from members of the public.
Miss Dando, 37, was killed by a single bullet to the head outside her home in Gowan Avenue, Fulham, south-west London, on 26 April last year.