His comments, made while on international duty, are known to have been greeted with disappointment and surprise by the club.
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.
Press Association Sport understands Fellaini will not be subject to sanctions when he returns to the club's Finch Farm training ground 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, which it is understood he does not deny saying during a press conference after training with his national side in Anderlecht on Saturday, by issuing a statement on his Facebook page.
"I was very surprised to read in some newspapers that I will leave Everton very soon and perhaps next January," he wrote.
"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."