Campbell is free to negotiate with a club after managing to wrangle his way out of a contract with League Two outfit Notts County.
A deal would not be able to take place until January, but reports suggest that United would like to invite the 35 year old to train with the club until a deal can go ahead.
Speaking to ESPN, Campbell revealed that he has not had any contact from United but would more than welcome a move.
"I have heard nothing officially from Manchester United, but there have been suggestions and of course, it would be a great move for me, if it happens.
"I shall just have to wait and see if a call comes through the right channels.
"Of course I would love to play for Manchester United, but I am not counting any chickens, just waiting to see what happens.
"I am training as usual at Arsenal, and would go to train at Manchester United if I was asked."