The 27-year-old former Manchester United target has joined Monaco on a one-year loan deal.
Monaco president Vadim Vasilyev said: “Coentrao arrives tonight in the form of a loan deal.
“Kurzawa wanted to go to PSG and we have been paid a very large amount. But we have found an alternative.”
Coentrao made just nine appearances for Real Madrid in La Liga last season.
A statement on Real Madrid’s official website added: “Real Madrid and Monaco have agreed to the transfer of player Fabio Coentrao until June 2016.”