However, Ferguson has confirmed he has no interest in a career in international football.
"No. I won't turn to international management," he told a special 200th anniversary edition of the club's magazine Inside United.
"When I've finished here, I think I deserve a rest. After here, I'm finished."
The involvement of a Great Britain side at the 2012 Games has yet to be confirmed and is opposed by the football associations of Scotland, Wales and Northern Ireland, who fear it could jeopardise their independent status within Fifa.
But just last month, when asked at a Downing Street press conference what he wanted for Christmas, Brown said: "That we have an Olympics team for the 2012 Olympics that is a football team from Britain."
Brown also suggested the involvement of Ferguson could help to bring the Scots on board, adding: "Sebastian Coe and I have talked to Alex Ferguson about the possibility that he might want to be the manager."
"I think there is a way around this problem, and I think the way around this problem can be found if people can get round the table and discuss it."
Earlier last year, however, Ferguson, 67, insisted that he was too old to guarantee his involvement.
"I'll be 70 then, so I don't need any commitment to anything like that," he said.
"I would not, in any way, shape or form try to commit myself to anything like that four years away.
"I'm not sure they would allow it (the GB Team) anyway, because countries have their own identity."