The Blues defender is set to be a free agent in under six months’ time, with his successful spell in West London due to end when his contract expires.
Having lost his starting role under Jose Mourinho, it appears unlikely that the 33-year-old will be offered a new deal, which has alerted Brendan Rodgers.
The Reds chief is keen to strengthen his squad in the next transfer window, with a new left-sided a defender a priority.
Jose Enrique’s absence has forced on-loan Aly Cissokho into the role, and the Frenchman has been erratic in recent weeks.
Rodgers worked with Cole during his time as a coach at Chelsea, and is known to be a big fan of the rampaging full-back.
However Liverpool may face competition from sides in the MLS.
The likes of the New York Red Bulls and Los Angeles Galaxy have been linked with Cole.
The American sides are understood to be ready to make him a lucrative offer to become one of their ‘designated players’ – as was the case when David Beckham moved stateside.