The Denmark international moved to Fener from Ligue 1 side Lille in the summer, but has attracted interest from the likes of Inter, Milan, Roma and Manchester United, as well as the French champions, as a result of his fine performances.
"Obviously it is very flattering to be mentioned in connection with other big clubs but I am not at all thinking about a move," the 26-year-old man.
"I am extremely happy in Fenerbahce, just like my family and I are happy with our life in Istanbul. And I really enjoy playing at a top club with ambitions to win every game, as well as winning the Turkish championship and doing well in Europe."
After failing to qualify for the Champions League group stage, Fener have since reached the last-32 of the Europa League and sit second in the Super Lig. The club also boast the division's best defensive record and Kjaer is confident his side can win silverware this season.
"Fenerbahce is a real top club when it comes to having a winning mentality," he continued. "We are only looking at our own performances and the fact that we want to be top of the league.
"We enter the pitch to win every game - and to become champions. I love these ambitions and that pressure.
"There is so much passion around Fenerbahce, and the reactions and support from the fans both in the stadium and on, for example, Twitter has been amazing."
Fener's final game before their mid-season break on December 28 saw them win 2-1 at home to Sivasspor and it proved to be a special occasion for the Dane as he scored the winning goal 13 minutes from the end – his first for the club.
"It was of course a fantastic feeling to score my first goal for Fenerbahce, which was also the match winning goal. I have come close to scoring a few times this season, so it was a good feeling to see the ball go in the net," he said.
"I am constantly working on improving my skills - including scoring more goals - but in Lille I took some of the direct free-kicks, and those gave me three goals last season. In Fenerbahce I don't take the free-kicks, so I can't score those goals here.
