Fener are currently in talks with the Blues but are determined to wrap up a deal for the player.
The former Crystal Palace star returned from a nasty knee injury last season but failed to find consistency in his 17 appearances under former boss Sam Allardyce.
The Turkish side are keen to wrap up several deals before the start of their Champions League play-off campaign, and new manager Phillip Cocu - who worked with Everton director of football Marcel Brands at PSV - looks keen on Bolasie.
A Spor believe former Besiktas forward Cenk Tosun has offered the 29-year-old advice on moving to his home country and has encouraged the move.
Fenerbahce are reportedly keen on a season-long loan deal.
Turkish outlet Takvim - via Sport Witness - believe Fenerbahce have upped their offer for Yannick Bolasie.
The Yellow Canaries were initially claimed to have offered the Blues £1.5m to take the winger on loan for the season, but the source now believes they’ve doubled their opening gambit.
Manager Phillip Cocu is said to be keen to work with Bolasie and former Liverpool sporting director Damien Comolli - who now operates in the same role for Fenerbahce - is leading negotiations with the Toffees.
The Turkish outfit are said to be keen on having a clause inserted into the deal which would allow them to retain Bolasie’s services on a permanent basis should he impress.