Sir Alex Ferguson appears to be the favourite to land the Dutchman ahead of the likes of Manchester City, Juventus and Paris Saint-Germain, who all now seem to have given up on the 29-year-old.
Gunners boss Arsene Wenger has already deemed one offer from United as too low, and Parlour believes that the Red Devils need to up their bid if they are serious about signing the Arsenal captain.
"It's a really awkward situation," Parlour told talkSPORT. "I'm sure Arsene Wenger wants to sort it out and say 'Robin, you're either staying or you're going.'
"Manchester United have been a little bit cheeky with a bid of say £15m because they know he wants to get away.
"Manchester City's interest has cooled a little bit and Juventus have cooled their interest. It just leaves Manchester United and they know he's desperate to leave so they're trying to nick him."
Van Persie returned to action with Arsenal as a second-half substitute against FC Koln on Sunday.