Speculation in recent days has linked Rooney with moves to Paris St Germain, Bayern Munich or Chelsea in the summer.
The 27-year-old forward was omitted from the squad for Ferguson's final game in charge at Old Trafford, a 2-1 win against Swansea, fuelling the rumours of an impending exit.
And Ferguson confirmed after the match that Rooney has asked to leave the club but indicated that he is still hopeful the England international will change his mind.
"I don't think Wayne was keen to play, simply because he's asked for a transfer," the Scot told Sky Sports.
"I think he wants to think things through in his own mind. I think that's a good idea. We're not going to let him go.
"I think he's maybe a bit frustrated at being taken off once or twice in the past few weeks."
This is the second time that Rooney has asked to leave Old Trafford, also requesting a move in October 2010 before agreeing a new deal with the club.
Sky Sports' Gary Neville says the striker needs to sort himself out.
"Wayne is an emotional lad and you can see in the last couple of months he hasn't quite been himself," said Neville.
"Wayne Rooney wears his heart on his sleeve. He fights every refereeing decision. But he has been a bit vacant on the pitch. You can see something is wrong.
"But there is nowhere to go from here. I have seen players join Real Madrid and other great clubs in Europe. But they always want to come back.
"If you play for this club for 20 years it is not a bed of roses. You struggle mentally at times. He will be going through a difficult moment.
"But in a couple of weeks he will be coming away with England. Then he needs to have the summer off and hopefully get his head back to being a Manchester United player."
Neville feels Rooney's family, and agent Paul Stretford, have a crucial role to play over the next few weeks.
"This is a good kid," said Neville. "He needs to surround himself with his family and his players. The ones who have been here for a long time will bring him round.
"He is not someone who will make rash decisions but he needs people around him to give him good advice. He has an agent and friends. They need to be giving him the right advice.
"This is the only club he should be playing at."
The comments came at the end of an emotional afternoon at Old Trafford with Ferguson urging Manchester United's fans to stand by David Moyes next season.
"I would like to remind you this club stood by me in bad times, the players and the staff," he said. "Your job now is to stand by the new manager."
Ferguson paid personal tribute to Paul Scholes, who started today's game and will also retire in the summer, and Darren Fletcher, who continues his recovery from a chronic bowel condition.
And he insisted he would still be a regular presence at Old Trafford.
"I will be able to go along and watch them rather than suffer with them," he said.
"Those last minute goals, the comebacks and even the defeats are part of this great football club. It has been an unbelievable experience."
Robin van Persie could not hide his delight to be involved in the occasion and added: "It is amazing. Only in my wildest dreams could I have imagined it would be like this.
"It is just a dream come true. We have got some fantastic players so hopefully we will win even more."
Asked about Ferguson's decision to retire, Van Persie added: "It was a really sad day. Everyone understands it and is very thankful to him.
"From my point of view, I think it is a massive honour to have worked with him for one year.
"Even though it was only one year and I haven't worked with him a lot, it has been an unbelievable year and he's made such an impact.
"He's a great manager and an even greater person. The way he is and the way he works with players, he deserves all the credit because he has been amazing."