The Blues have had a bid accepted for the sought after youngster.
The defender is out of contract in the summer, and has told the club he will not sign a new deal, leaving chairman Massimo Cellino to accept the bid from the Toffees rather than risk a lower fee at a tribunal, as he is under 24-years-old.
Roberto Martinez’s side have beat off stern competition in the Premier League as well, according to the report from the Express, with Aston Villa, Newcastle United and West Ham all interested in his services.
Those three clubs had offered a similar price for Byram, but he prefers a move to Merseyside given Martinez’s track record of developing young talent.
The Catalan will hope that the young defender can come in and provide competition for Seamus Coleman, with them lacking cover for the Republic of Ireland international, who is currently out injured.
He will become their second signing of the January transfer window, after they also completed a deal for Swiss youngster Shani Tarashaj last week.