The midfielder, 28, had been expected to join Arsenal.
But now it seems Gunners boss Arsene Wenger is set to lose out to Old Trafford boss Alex Ferguson in the race for Cole's signature.
Spurs boss Harry Redknapp and Real Madrid chief Jose Mourinho have both managed Cole in the past and are keen to do so again - but are also set to miss out.
Chelsea skipper John Terry admits he is stunned at the move.
Terry said: "It still amazes me how the club have come to the conclusion of letting Joe go."
The feeling at Old Trafford is that a lot of the speculation surrounding United's reported interest has been generated by people close to Cole in the hope of accelerating interest from other clubs.
Cole's future now looks like a straight choice between Arsenal and Tottenham, the north London rivals having both initiated talks with the player's representatives.