Sky Bet suggests United are 2/5 to complete the signing before the winter deadline - the best odds they're offering on a particular player to join the defending Premier League champions.
United boss David Moyes has been linked with a number of targets this month. And while central midfield is thought to be a priority, the club is also thought to be chasing cover at left-back.
But Coentrao, who was tipped to move to United on loan at the end of last summer, is the most likely addition.
The Real Madrid star has struggled for first-team action at the Bernabeu under Carlo Ancelotti, although the former-Chelsea boss has publicly said he wants him to stay.
The Portugal international played in the Copa Del Rey on Wednesday night but was sent off after 85 minutes for two yellow cards.
Moyes is under pressure to make signings this month following a disappointing start to his first season in charge at Old Trafford.
The reigning Premier League champions currently sit seventh in the table.