Lawrenson, who most regularly works for the BBC's Match of the Day programme, made the astonishing claim on Saturday morning as a guest on the Today FM radio show in Ireland—citing the "whispers" of those journalists who cover the club on a daily basis.
In quotes picked up by the Daily Mirror, Lawrenson said:
There's all sorts of rumours coming out of the training ground at the moment.
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 [United manager] David Moyes.
While it seems unlikely that, if the reports were true, the story would not have already made it into the mainstream press—if journalists had picked up on the "whisper" on Tuesday, they will have done all they could to substantiate it enough to publish something—it is not implausible that van Persie might be unsettled at Old Trafford.
Almost ever since Moyes' arrival at the club there have been reports, via Mark Ogden of the Telegraph, of van Persie's concerns at his training regime, with the Dutchman—who had a history of injury problems during his time at Arsenal— believing he is not being handled as carefully as he was during his first season at United under Sir Alex Ferguson and has thus become more susceptible to injuries.
In June, former Wales fitness coach Raymond Verheijen described Moyes' methods as "prehistoric" after van Persie picked up an injury on the club's preseason tour.
"In Sydney, Moyes said: 'We’ve overtrained Robin this week to build up his fitness'. [In Japan] he picked up a muscle injury," Verheijen said on Twitter, as reported by the Daily Mail. "It really makes you wonder how these prehistoric training methods can still take place at the highest level.
"The only way to solve this problem in Jurassic Park is to improve education of these dinosaur coaches, fitness clowns & scientific cowboys."
While the veracity of such claims—and the links between Moyes' methods and van Persie's injury—are hard to ascertain, van Persie has been sidelined more often this term than he was last season, most recently missing three successive games with what Moyes would only describe curtly on Wednesday evening as a "normal groin injury."
On Thursday, he added to the club's official website: "I don't know [when he will return]. We've been trying to get him back for a few weeks.
"But he is not ready yet. It's his groin. It's a sore groin injury and nothing desperately serious."
There have also been reports previously of a falling out between the pair after van Persie was left out of a Champions League squad to face Real Sociedad, suggesting the relationship between player and manager is not as close as it was when Ferguson was at the helm.
Whether that rocky relationship with his manager is enough for van Persie, who was only recently expressing his desire to sign a new contract with the Premier League champions, to ask to leave is a different matter, however.
Moyes is also facing issues with another player after Rio Ferdinand publicly criticised his method of not telling players whether they are in the team until the day of a match.
Ferdinand told BT Sport, via the Daily Mail:
Clive Brunskill/Getty Images
This manager is a bit different in that he doesn't name the team beforehand, you don't get to know the team.
The old manager used to give you a little bit of an idea if you were playing. When you know you're playing the intensity goes up a little bit.
That's what you've got to try and do even if you're not sure if you're playing. It's hard to do mentally. You spend a lot of nervous energy thinking 'am I playing, am I not playing'.
Keep just going round in circles in your head, enough to turn you into a mad man.
After an underwhelming start to the season that sees United 12 points behind league leaders Arsenal, Moyes will want a win on Saturday against Newcastle United to steady the ship some what.