The experienced full-back has fallen out of favour at Stamford Bridge since picking up a rib injury back in November, and has lost his place to right-back Cesar Azpilicueta.
The Chelsea veteran's problems were exacerbated when he was spotted attending former club Arsenal's Christmas party, after Chelsea boss Jose Mourinho had cancelled his side's event.
Mourinho then insisted he wouldn't feature in the 0-0 draw with Arsenal last Monday - a threat he went through with - although he did recall Cole for the 1-0 over Swansea yesterday.
According to the Express, enough damage has been done to Cole and Mourinho's relationship that the former, who signed a one-year contract extension in January, will be allowed to leave after seven years of service.
The paper reports that both Tottenham and Liverpool are in the running for his signature - although a move is unlikely to happen in January.
MLS outfit New York Red Bulls are also said to have offered him a deal, while previous reports have suggested that the 33-year-old is considering retirement.
Spurs look to be short of options at left back and were forced to field centre-back Jan Vertonghen there while regular starter Danny Rose was out of action - although whether the former Arsenal player is willing to move to White Hart Lane remains to be seen.
Liverpool meanwhile have Jose Enrique and Aly Cissokho as their man options at left-back, although recently youngster Jon Flanagan has been the main man for boss Brendan Rodgers.
Chelsea are likely to be unwilling to sell to either Liverpool or Tottenham - although Cole will be free to discuss offers at the end of the season, leaving the west London club powerless to stop him from joining either club.