The Dutch striker, who has been sidelined in recent weeks with a combination of groin and toe injuries, is said to have informed manager David Moyes of his desire to leave Old Trafford on Tuesday.
"There's all sorts of rumours coming out of the training ground at the moment," Lawrenson told Today FM. "One or two of the press boys that I know that cover Manchester United, they're even saying that the big whisper is Van Persie might have asked for a transfer on Tuesday. So if that happens, what a mess that's going to be for David Moyes."
Van Persie joined United last summer from Arsenal and he went on to score 26 Premier League goals as the Red Devils won the title for a 20th time.