David Moyes came close to sealing a loan deal for the Portuguese ace last year, but saw his efforts to get him in thwarted as the transfer deadline passed.
They have since kept an eye on Coentrao, who has struggled for game time all season at the Bernabeu, managing just six league outings in the white jersey.
Although Southampton’s Luke Shaw remains the top target to replace Patrice Evra at left-back, the 26-year-old is seen as a viable and cost-effective alternative.
Coentrao is able to play in a defensive role or in a more advanced position, and has long been hailed as a top talent after emerging at Benfica.
Chelsea have also been linked with him, but with Cesar Azpilicueta playing well on the left side of the Blues’ backline Jose Mourinho’s interest has cooled.
Real are said to be willing to accept around £10m for their man as they look to clear space for their own summer arrivals.