Officers suspect Kelly Buckingham, from Lincolnshire, was assaulted about the time she was last seen in November.
But they said friends may mistrust the police because she was wanted after failing to appear in court in August.
The reward was issued on her 47th birthday to help find Ms Buckingham, from Holton le Clay near Grimsby.
Det Insp Richard Myszczyszyn from Lincolnshire Police said it was unusual to offer a reward for help.
"Kelly's friends and acquaintances may be distrustful of police and may not believe us now when we say that she is being treated as a missing person," he said.
"But I would like to assure them that our concern for Kelly is genuine and our efforts to find her are very much driven by our fear that something has happened to her."
Police have taken the "unusual step" of offering a £1,000 reward in the search for a woman who has been missing for more than a month.