According to Mirror.co.uk the Dutchman is reportedly unsettled under new boss David Moyes and could be looking for a way out of Old Trafford.
Speaking on Today FM in Ireland, Lawrenson said he had heard numerous reports that last season's Premier League top scorer could be angling for a move away.
"There's all sorts of rumours coming out of the training ground at the moment," he said.
"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."
It would certainly be a big blow for the Scot as he looks to reinvigorate his side after a slow start to the season, the latest setback being Wednesday night's disappointing home defeat to Everton.
The Red Devils host Newcastle today where van Persie could return for the first time since netting the winner against former club Arsenal on November 10.
Image: Robin van Persie and David Moyes
Photograph: Alex Livesey/Getty Images
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