Coentrao was set to join the Premier League champions on transfer deadline day, only for the deal to collapse with minutes to spare before the window slammed shut.
The Portugal international will now remain in the Spanish capital and is set to fight Marcelo for a first team place under new coach 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 los Blancos from Benfica for €30million, 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 thrashing of Australia.