The Daily Mirror suggest the north-west duo are interested in signing Cole on a free transfer if he decides to leave Stamford Bridge at the end of the season.
Cole has struggled for games this season as Jose Mourinho has tended to deploy Cesar Azpilicueta on the left-hand side of defence.
Chelsea are said to be willing to offer Cole a new short-term deal, however the former Arsenal star may opt to leave the west London outfit in order to play regular first-team football.
Several top Premier League clubs are in the running to sign the 33-year-old England international - with Liverpool and Real Madrid said to be vying for his services.
However the aforementioned duo will face stiff competition from both Manchester United and Manchester City if they do look to stiffen up their interest in Cole.
Manchester United will be in the market for a new left-back in the summer as Patrice Evra has yet to sign a new deal with the Premier League champions.
The Red Devils will need another left-back even if Evra does stay as Alexander Buttner has failed to challenge the Frenchman for a starting berth.
Manchester City, on the other hand, could be inclined to sell Aleksandar Kolarov to Juventus as he is out of contract next summer.