The Croatia international was heavily linked with an exit in the summer of 2017.
Premier League clubs were reported to be keen on taking the 24-year-old midfielder, with fierce competition for places leaving him scrapping for minutes in the Spanish capital.
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Kovacic was, however, convinced to stay put after speaking with Zinedine Zidane, with the Blancos having invested €29 million (£25m/$34m) in his ability when luring him away from Inter in 2015.
Quizzed on how close he has come to leaving Madrid by 24sata, Kovacic said: “Offers last summer? If Zidane tells you to stay, that you will play and that he has faith in you - you simply stay! It is Zidane!
“It was a privilege to hear such words from someone like him, who was one of best ever. And also, it is Real Madrid... how could I leave Real?! It is an honour to be part of this club.”
He added: “I am still young and have a lot to learn.
“To be honest, when Zidane promised us at the beginning of the season that we will all get minutes, I thought it would be hardly doable with so many of us in the squad. But he did it, we all got a chance. It is pure pleasure to play under Zidane.”
Kovacic has made over 100 appearances for Madrid across three seasons, with his ability to settle in Spain aided by the presence of fellow countryman and perfect role model Luka Modric.
“Modric and [Andres] Iniesta are the best midfielders I have seen. I was watching them play and wanted to play like them,” added a man going in search of a third Champions League crown against Liverpool on Saturday.
“When I came to Madrid, Modric didn't give me any football advice because he kept telling me I was ready to play for Real.
“But he gave me some tips about life, like: ‘Don't go crazy, you have to be calm and cool’. I just followed his example. And that is why I am still at Real.
“Modric is undoubtedly the best Croatian player ever. He is also an amazing guy and I am proud to have him as a friend.”