According to the Mirror, the Chelsea and England left back is facing an uncertain future for club and country.
Red Bulls have a vacancy for one of their three designated 'marquee' players and know that Cole has always been keen to end his career in Major League Soccer, the report said.
They want to link Cole up with his former Arsenal teammate Thierry Henry in the January window before the new MLS season starts in early March, the report added.
LA Galaxy are also keen to sign Cole, who has won 106 England caps but is now facing the prospect of being frozen out of Roy Hodgson's World Cup squad, the report further said.
Cole, who turns 33 this month, is out of contract at the Stamford Bridge club in the summer and is unlikely to be offered a new deal, according to the report. (ANI )