Coentrao was set to join the Premier League champions on a season-long loan deal at Old Trafford on transfer deadline day, only for the deal to fall through at the last minute.
The Portugal international will now remain in the Spanish capital and is set to fight Marcelo for a first-team place under new manager Carlo Ancelotti.
And Coentrao's hopes of playing regular first-team football received a boost on Saturday when Marcelo suffered a hamstring injury in Brazil's 6-0 friendly thrashing of Australia.
Coentrao told O Jogo: "I want to do better than last season, despite already being at a pretty high level.
"I want to show everyone that I deserve to be at Real Madrid.
"There was this chance to go to Manchester United, but I am at Real Madrid and that's what matters."
Despite joining Real from Benfica for a €30million fee, Coentrao has struggled to make himself a regular at the club.
The 25-year-old has made just 63 appearances in his two seasons at Real, and is yet to feature in the current campaign, but that is likely to change when they travel to Villarreal on September 14.
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