The west London club had a second bid for the 27-year-old rejected on Monday, and the former Everton forward, who will miss United's pre-season trip to Sweden with a shoulder injury, has previously expressed his desire to leave.
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"You do worry he'll leave because he's a fantastic football player, but you leave that to Wayne and the club," Fletcher told Sky Sports.
"I hope he stays because he's been a fantastic player for Manchester United, and a big part [of their success].
"I hope he stays and achieves more things with Manchester United."
"In all he's achieved in the last 9/10 years since I've been in the squad, we've been in the squad together, so I hope he stays and achieves more things with Manchester United."