The 33-year-old has made 338 appearances for Chelsea since joining the club from Arsenal in 2006, helping them to win the Champions League in 2012 and the Premier League in 2010.
The Chelsea full-back confirmed on Twitter earlier this week that he is unlikely to sign a new deal to extend his stay at Stamford Bridge beyond the summer - and is now being pursued by up to eight clubs.
According to Gazzetta dello Sport, Cole is set to snub interest from the Premier League and the likes of Real Madrid, and head to France in order to join Monaco.
The Italian source claims that the Chelsea defender has reached an agreement with the Ligue 1 side, and will join when his current contract expires.
That will come as bad news for the the likes of Tottenham and Manchester City, who had been linked with the full-back.
Chelsea are expected to pursue Sevilla youngster Alberto Moreno as Cole's long-term replacement this summer.