The Portuguese defender looks set to leave Los Blancos after a difficult season in which he has only made seven La Liga appearances.
Coentrao came close to joining Manchester United on loan last year, yet the deal fell through, but he is rumoured to be more than willing to make a switch to the Premier League this time around.
Liverpool have now emerged as prime candidates for his signature, with the Merseysiders understood to have been in contact with his agent, Jorge Mendes.
The Reds are said to have discussed a potential deal, and plans for a bid are at an advanced stage.
City are also keen, but it is unclear as to whether or not the 26-year-old is a top target.
Real are set to demand around £11.5m for their man, which is a figure Brendan Rodgers may be willing to stretch to in a bid to solve his current left-back problems.
Tottenham could also revive their interest after baulking at his £15m valuation in 2013.
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