The 29-year-old, who has been forced to take a break from football due to the bowel condition ulcerative colitis, has played alongside Rooney for each of the last nine seasons and rates his team-mate as 'a fantastic player'.
Fletcher has not spoken to Rooney about his future - Chelsea have had two bids for the England striker rejected by United - but admits he does not want him to leave the Premier League champions.
"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 News.
"I hope he stays because he's been a fantastic player for Manchester United, and a big part (of their success).
"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."
Fletcher also revealed that he is keen to see United transfer target Cesc Fabregas make the move to Manchester from Barcelona - but has backed new boss David Moyes to be a success no matter who the club sign this summer.
"He (Fabregas) is a fantastic player, we're always linked with great players," he added. "Whether that's going to happen or not remains to be seen, but I think it's part and parcel of being at the club. We're always going to attract and want to sign big players that will add to the fantastic squad we've already got.
"I think even if we don't sign anyone we have the capabilities and the squad to achieve the success that we want to achieve this season. You're talking about English champions so it can't be too much of a concern."
Meanwhile, Fletcher is unsure when he will return from his protracted absence having undergone numerous operations - the last of which took place in January - and been put on a strict dietary regime in a bid to correct his stomach condition.
The 29-year-old will miss the start of the season, and Fletcher admits he has no firm date pencilled in for his comeback.
Fletcher admitted: "I'm still in the hands of the surgeons at the moment - I've not actually started training yet.
"I've been doing my own little bit in the gym, which is keeping me ticking along nicely and I'm waiting for the surgeon to give me the go-ahead to join in training.
"I don't have a timescale for when I will be back playing really. Hopefully it is not going to be too far away in the distant future.
"I just need to be patient. After all I've gone through I think it's important that I'm patient. If it means a few extra weeks to get myself right, then so be it."