Manager Sam Allardyce wants to make the 26-year-old his first major signing – in a deal worth around £25m – since succeeding Ronald Koeman.
Allardyce is also pondering a £25m move for Sevilla midfielder Steven Nzonzi, who played for him at Blackburn Rovers.
Tosun, who has been linked with Crystal Palace and Newcastle, helped Besiktas reach the Champions League last 16, where they will face Bayern Munich.
He scored his ninth goal of the season on Thursday in Besiktas’ 4-1 win over Osmanlispor.
No deal has yet been finalised, but Everton hope to conclude business swiftly once the transfer window opens on 1 January.
The failure to replace Romelu Lukaku after his summer move to Manchester United has left the Goodison Park club short in attack.
However, Everton major shareholder Farhad Moshiri is willing to back his new manager in January and a left-back is also a priority, with Leighton Baines injured.
Allardyce is unbeaten in his seven games in charge, with one win in the Europa League and three wins and three draws in the Premier League moving them up to ninth place.