The Daily Mirror reports the Blues are preparing a $35.5 million (US) bid for the 24-year-old.
Fellaini has spent four seasons at Everton since signing with the club in 2008.
His afroed silhouette featured for David Moyes immediately, and the player has become a cult hero at Goodison Park in his 120-plus appearances.
This season, the Belgian international has played 37 times and scored six goals for Everton.
But Fellaini's role cannot be summed up in goals scored. He is charged with being the disruptor at the heart of Moyes' team while at the same time linking play from his deep midfield role.
He leads the team (and is top 10 in the Premier League) in tackles per game, and no one in the EPL commits more fouls per game than Fellaini (many of them tactical). But he also leads the Toffees in passes per game.
Despite Roberto Di Matteo's late-season rescue act incorporating most of the veteran Chelsea stars, Abramovich appears to be set on turning the squad over in the summer.