The Sun says Cole has told pals he wants to join Manchester United.
The England ace becomes a free agent next Wednesday when his Chelsea contract expires.
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.
Terry said: "It still amazes me how the club have come to the conclusion of letting Joe go."