The Belgium international has been the driving force behind
However was quoted by Het Nieuwsblad as saying: "I have started my fifth season at Everton. This will be one of my last seasons. I have seen everything. In January, or at the end of the season, I will look for another club or league."
His comments, made while on international duty, are known to have been greeted with disappointment and surprise by the club.
But sources insist the 24-year-old has never given any indication of angling for a move having signed a lucrative five-year contract last November.
The midfielder will not be subject to sanctions when he returns to the Everton training later this week, although he may have to explain himself to manager David Moyes, and club officials hope the Belgian will be wiser for the experience.
Fellaini sought to diffuse the comments although he does not deny saying them during a press conference after training with his national side on Saturday.
"I was very surprised to read in some newspapers that I will leave Everton very soon and perhaps next January," he wrote on his Facebook page.
"Let me remind you that I am very happy in Liverpool and our start of the championship was good. Kevin Mirallas and some other good players joined us to make a better squad.
"So with Everton, I am ready to play all the season and to reach our goals of this season."