Atletico lost 3-2 on aggregate to Juventus in the last 16 on Tuesday, a defeat made worse by the fact the tournament's final will be played at their own Wanda Metropolitano in June.
Simeone, who signed a new contract in February until 2022, has turned Atletico into one of the most feared teams in Europe since he took charge eight years ago.
But there has been criticism of his defensive style of play while his team are set to end this season without a trophy, given they also lie seven points behind Barcelona in La Liga.
"I fully trust in our project and in Simeone," said Gil, in an official club statement on Friday.
"I offered Diego his new contract the week after the 4-0 defeat in Dortmund and it was not to go against the grain or to thank him for these seven years.
"It was because I am convinced that we will enjoy the next three years just as much as we have done so far. I'm sure Diego will complete a decade at Atletico Madrid that will change our history."