Robson told PA Sport: "It wouldn't surprise me if he was offered a coaching job there.
"He's done a couple of training badges and Sir Alex is encouraging that sort of system at the moment at United.
"Ryan is a determined character so it wouldn't surprise me if they did offer him a role somewhere along the line as coach.
"He's definitely got the personal qualities to do the job and he's Manchester United through and through.
"With his record at United and the way he grew up with the club, he'll have no problems with doing that job."
Robson added: "Ryan deserves a new contract the way he has started this season. He's only played a few games but when he has played he's been really sharp and bright about his game.
"When I have been in to see them at training he's been looking really good so it doesn't really surprise me that United are talking about offering him a new deal."