The Southampton defender has been the subject of intense transfer speculation in recent months, with suggestions that he could leave St Mary's at the end of the season.
Saints boss Mauricio Pochettino has frequently denied that Shaw is set to be sold, reiterating that the England U21 international is happy on the south coast, after signing a bumper five-year contract back in August.
David Moyes has been credited with an interest in bringing the 18-year-old to Old Trafford, as he continues his search for a long-term successor for Manchester United vice-captain Patrice Evra.
The Red Devils manager has also been linked with a move for Leighton Baines - who he worked with during his time at Everton - but after failing to lure him from Goodison Park, is now understood to have turned his attention to Shaw.
However, Moyes could face competition for the talented teenager's signature from rival suitors Chelsea, particularly after Cole's Twitter comments encouraging Jose Mourinho to make a bid.
"How does it feel that Chelsea want to buy Luke Shaw to play over you?" one follower asked the 32-year-old during a recent Q&A session. "Hope to see him there to be honest" was his response.
Cole's stance might come as a surprise, particularly at a time where the full-back finds himself out of first-team favour in west London, with Mourinho currently preferring to play Cesar Azpilicueta out of position on the left side of defence.