Coentrao, 27, joined Real from Benfica in 2011 for a fee of 30 million euros (£22 million).
However, he is behind Brazil fullback Marcelo in the Real pecking order and has also been hampered by a series of injuries, making only 17 appearances in all competitions last season.
Monaco failed to qualify for the Champions League group stage when they were beaten 4-3 on aggregate by Valencia despite winning Tuesday's playoff second leg 2-1.
(Reporting by Iain Rogers, editing by Ed Osmond)
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