The 28-year-old playmaker, who is currently training with England ahead of the World Cup finals in South Africa, is widely expected to depart Stamford Bridge on a free transfer having yet to agree a new deal with Carlo Ancelotti's side.
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The former West Ham starlet struggled to cement a regular place in the Italian's starting line-up following eight months out through cruciate knee ligament damage, with reports suggesting that both Arsene Wenger and Sir Alex Ferguson are keen to recruit the trickster.
Tottenham Hotspur boss Harry Redknapp recently hinted that Cole - having made more than 200 appearances for Chelsea - has already agreed a deal with a club, fuelling speculation that he will be plying his trade at either Emirates Stadium or Old Trafford next season.
"It looks to me someone's going to make a great free transfer," he said. I've got a feeling that he's already agreed a deal with someone. The feedback we've had is that he might well have already done something."
And former World Cup winner Desailly, who made 156 league outings for the Blues following a £4.6 million move from AC Milan in 1998, revealed that - at this stage of his career - Cole would benefit more by being part of an experienced Red Devils side than a youthful Gunners team.
Speaking exclusively to FourFourTwo.com, he said: “A few years ago I would have thought he would go to Arsenal, but now that he is 28 he needs to go to a team that – with him or without him – will succeed, in order to put pressure on him to deliver performances.
“At Arsenal there are younger players and you need to talk to them, surround them and have a positive attitude around them.
“In Joe’s current situation he needs to push on. He doesn’t need to help the others, he needs a club to help him discover his best levels once again. Therefore, I would say Manchester United.”
FourFourTwo.com caught up with Desailly at Xbox LIVE Game with Fame, where he played 2010 FIFA World Cup South Africa against members of the Xbox LIVE community to celebrate the launch of the game and support the charity 1Goal.
