The Croatia international linked up with the Blues early in the 2018-19 campaign on a season-long loan.
He had been deemed surplus to requirements at the Santiago Bernabeu amid fierce competition for places.
Those at Stamford Bridge have embraced his presence, with more combative and ball-playing qualities added to the Chelsea engine room under Maurizio Sarri.
A permanent switch is now being mooted and Kovacic is open to a move if all parties are in agreement, telling Sky Sports: "I think it [a loan switch] was the most important thing I did for now in my career.
"It was a step I wanted a lot, I wanted to come here. I say thank you to my previous club, to Chelsea as well, who wanted me a lot. I am enjoying it and I like it a lot here.
"For now, I am happy here. The city is amazing, the club is one of the best in the world, and my team-mates are great.
"I need to respect my previous club, Real Madrid. I can imagine myself staying here, but I have a contract I need to respect, and that is what I will do.
"For now, my business is to give my best every day."
While Kovacic is being linked with a possible transfer away from Real Madrid, a current club colleague is being talked up for a big-money move in the opposite direction.
Speculation regarding Eden Hazard’s future refuses to go away, and the Belgian has asked about life in the Spanish capital.
Kovacic added on a talented team-mate: "For me, he is an incredible player.
"I did not know he was that good and he can easily be one of the two or three best players in the world.
"He asked me how it was there [at Real] but we are looking forward to having a great season together, to be fully concentrated on Chelsea.
"He is focused a lot on doing great things with Chelsea."
Chelsea face a trip to Watford on Boxing Day, with the Blues looking to bounce back from a shock defeat to Leicester and cement a standing inside the Premier League’s top four.