The London Evening Standard reports that Villas-Boas is keen to sign the defender on-loan in January.
The £100,000-a-week defender has started only three games this season for Real Madrid and could be interested in leaving the club.
Coentrao came close to joining Manchester United on loan on the final day of the transfer window, but the two clubs failed to reach an agreement in time.
Reports claim United could also make a new attempt at signing Coentrao to a permanent deal in January.
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Liverpool have unveiled their new home kit for next season, sticking with the hugely popular retro styling of the current shirt.
The reputation of the Premier League means it is the most popular and one of the most watched leagues in the world.