The Croatia international was among those to head for the exits at the Santiago Bernabeu during a summer of change, with a short-term deal taking him to Stamford Bridge.
Maurizio Sarri has embraced the qualities he brings to the Blues and made him a regular presence in a midfield unit which also includes N’Golo Kante and Jorginho.
Kovacic admits to enjoying his time in England, but is reluctant to be drawn on whether his time with Chelsea could be extended beyond the end of the 2018-19 campaign.
Quizzed on his future, the 24-year-old said in the Evening Standard: "It’s too early for these questions, I am here and very happy.
"[Playing regularly] is good, I’m a little bit tired. I was not used to playing every three or four games and here it is very tough football.
"You can’t relax against anybody so it’s a little bit difficult but I really enjoy it. We have a great team, great fans and the city is nice. I like it a lot here."
Kovacic added on his experiences in west London: "I go around and I enjoy the city - it’s huge and I haven’t seen enough of it yet. I’m looking forward to getting to know it more. I’m enjoying it. What’s the most interesting thing I’ve seen so far? The stadium.
"I don’t do anything special. I spend a lot of time in Cobham because I need to prepare myself for training and every game but after that I go home and enjoy time with my wife.
"Nothing special, we go for dinner, we live in the city so we are walking a lot. My family enjoys the city and that’s the most important thing for me in the end."
Kovacic has taken in 10 appearances for Chelsea across all competitions, including eight in the Premier League.
Having signed a few days after a Community Shield to defeat to Manchester City, he is yet to come unstuck as a Blues player.
Sarri’s side are currently on a 12-game unbeaten run and will be seeking to stretch that when then they take in a trip to Burnley on Sunday.