The 25-year-old was set to join the Premier League champions on transfer deadline day, only for the deal to collapse due to the paperwork not being completed in time to ratify the agreement.
The Portugal international will now remain in the Spanish capital and is set to fight Marcelo for a first-team place under Carlo Ancelotti.
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 Madrid from Benfica for €30 million, Coentrao has struggled to make himself a regular at the club.
The 25-year-old has made just 63 appearances in his two seasons in Madrid and is yet to feature this season, but that is likely to change when they travel to Villarreal on September 14 after Marcelo suffered a hamstring injury in Brazil's 6-0 win over Australia on Saturday.