The Premier League champions were close to bringing the Portugal full-back to Old Trafford on transfer deadline day, only for Real to pull the plug on the deal.
The 25-year-old was reportedly upset when the move to United did not go through, but he now insists that he only wants to prove that he deserves to play for Real.
"There was the possibility of joining Manchester United, but I stayed at Real Madrid, and that's what matters," Coentrao told reporters.
"It would have been bad if I had to stay despite having the chance to move to a better club, but that didn't happen as I already am at the best club in the world.
"I want to do better than I did last season, although I was at a very high level already. I want to show everyone that I deserve to be at Real Madrid."
Coentrao moved to the Santiago Bernabeu from Benfica in 2011.