The club tweeted a "thinking face" emoji alongside images of various Spanish newspaper reports detailing the admission on Thursday.
Clattenburg's comments have made headlines in Spain, where Atletico fans have long felt that Sergio Ramos' 15th minute goal -- in a game that Real Madrid went on to win 5-3 on penalties -- should not have stood.
"Real Madrid went 1-0 up in the first half, but the goal was slightly offside and we realised at half-time -- it was a hard call and my assistant missed it," the British referee told the Daily Mail. "I gave Atletico a penalty early in the second half. Pepe was furious and said to me in perfect English: 'Never a penalty, Mark.' I said to him: 'Your first goal shouldn't have stood.' It shut him up.
"People might think that sounds odd, because two wrongs don't make a right and referees don't think like that, but players do," Clattenburg went on. "I knew by saying that to him it would make him more accepting of the situation. He was a wind-up merchant and not fun to referee one bit, you had to be on your guard constantly."
Real Madrid's 2016 victory was their 11th European Cup and their second derby win in a Champions League final.
They had also beaten Atletico in Lisbon two years earlier, a game in which Ramos scored a dramatic late equaliser to take the match to extra time.
Real Madrid followed their triumph in Milan by lifting the Champions League trophy in Cardiff in 2017 and Kiev in 2018, becoming the first team to win it in three consecutive years in the modern era.
Atletico, beaten in all three of their final appearances, are still yet to win European football's biggest prize.