Rooney’s future remains uncertain and the striker could potentially play at Old Trafford for the final time this Sunday for Michael Carrick’s testimonial.
His boyhood club Everton are in the box-seat to sign him should he choose to leave United.
But Robson, United’s legendary longest-ever serving skipper, believes the club’s record goalscorer should stay and fight for his place.
“I‘ve heard Mourinho wants him to stay,” said Robson, speaking on behalf of United’s global betting partner Marathonbet.
“It will cost a lot of money to bring two new strikers into the club, to go with [Marcus] Rashford and [Anthony] Martial.
“So I would like Wayne to stay. He hasn’t had a great season but he still has unbelievable ability.
“It comes down to how much desire he has got and how much he wants to break more records and keep going at United.
“Only Wayne can make that kind of decision.”
If Rooney does bid farewell, Robson insists world record signing Pogba is tailor-made to take over the armband duties.
Pogo hasn’t been short of critics following his £86million return to the club last summer.
Yet Robson, 60, insists he has all the hallmarks of what it takes to lead United.
“I think Paul Pogba could be a captain for Manchester United, where he leads by example,” he said.
“I’ve got to say I’ve liked Ander Herrera’s attitude. I think he’s took it all board, trying to organise people and gee them up and gee the fans up when things haven’t been going so well. Antonio Valencia has got that determination about him too.
“But with Paul, I still think that he has improved us.
“I think whenever he’s been missing, we’ve missed a little something.
“Perhaps sometimes he does dwell on the ball a little bit too much for a central midfield player and you want him to move it a little bit quicker. But Paul shows his strength and he has got real ability.
“I still feel like he’s going to improve again next year.”
Bryan Robson was speaking on behalf of Marathonbet, the Official Global Betting Partner to Manchester United