The England left back is in the final year of his deal at Stamford Bridge, and Chelsea owner Roman Abramovich recently stated players over 30 would only be granted one-year extensions.
Cole is believed to want a longer term deal, and with no talks about an extension scheduled, is free to negotiate with foreign clubs from January onwards.
Reports suggest Cole's former-Blues boss Jose Mourinho is monitoring the situation, and would be interested in bringing the defender to Real Madrid.
But the Portuguese boss faces competition from another former Stamford Bridge manager. PSG boss Carlo Ancelotti is a keen admirer of the left-back, and representatives from the French club will reportedly offer Cole a big-money offer in January.
The 31-year-old is thought to have begun discussions over a new deal at Stamford Bridge, but Roman Abramovich's refusal to offer longer-term deals to senior players may force Cole out.
PSG, who second in Ligue 1 table behind Marseille, have been one of the biggest spenders in world football over the last six months, bringing in AC Milan duo Thiago Silva and Zlatan Ibrahimovic, and signing an agreement to bring highly-rated Brazilian midfielder Lucas Moura to the French capital.
Do YOU want to write for GiveMeSport? Get started today by signing-up and submitting an article HERE: https://www.givemesport.com/writeforgms