The 24-year-old Belgian international has been in scintillating form for the Toffees this season, and once again played a starring role in yesterday's 2-1 victory over Sunderland.
Fellaini scored his sixth goal of the campaign to draw David Moyes' side level, and set up Nikica Jelavic for the winner in Saturday's Premier League clash at Goodison Park.
Blues boss Roberto di Matteo wants to add to his attacking options at Stamford Bridge, providing support for Fernando Torres, who continues to lead the forward line.
Roman Abramovich is believed to have given the green light to a big-money bid for Fellaini in the New Year, with the fact he would be eligible to play for Chelsea in the Champions League another attractive proposition for the Russian owner.
The 6ft 4in forward insists he's happy on Merseyside, but recently revealed his desire to play in Europe's elite club competition, and intimated he may be forced to leave in order to fulfil his ambition.
A move to west London would see Fellaini reunited with international team-mate Eden Hazard, who has been a revelation since his £32million move to England from Ligue 1 side Lille in the summer.