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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Since Guus Hiddink arrived at the helm, Chelsea have improved considerably after a tricky start to the campaign.
Newcastle United are staring down the barrel of relegation to the Championship for the second time in the past eight years.