According to The Guardian, the club have agreed a £25m fee with Besiktas for the 26-year-old while The Times claim that the Blues have offered him a contract and are negotiating a deal worth £20m in the hope that he would complete his move when the transfer window opens in the coming days.
Sources in Turkey, meanwhile, are suggesting that Tosun has already agreed terms on a four-and-a-half-year deal with Everton at Finch Farm which would suggest that the only thing standing in the way of the transfer is final agreement with Besiktas.
He was in action for Besiktas this evening, scoring against Osmanlıspor in a 4-1 win which the Turkish Football website was billing as his farewell game before he signs for Everton. It certainly looked that way as he appeared to bid farewell to the fans after being taken off in the second half and then again at the final whistle.
If true, it would represent victory for the Blues in a battle with Crystal Palace for his signature. Sam Allardyce is said to have recommended Tosun to the Eagles' hierarchy when he was manager there last season and seems to have had the same advice for Steve Walsh at Goodison.
Tosun has scored eight times in the Turkish league this season, the same number as former Blues forward Arouna Kone has managed for Sivasspor, while also adding four more in six Champions League fixtures.
One of them, a 30-yard rocket against FC Porto, was named goal of the group stages: