The Toffees have been trying to negotiate a deal since the transfer window opened earlier this week and it is understood a fee of £27million has been agreed.
Tosun was already waiting in London as talks continued and will now travel to Merseyside to discuss personal terms before undergoing a medical.
Allardyce confirmed the deal is nearing a conclusion and revealed the striker could be available for Friday’s FA Cup third-round clash with Liverpool.
“We’ve agreed a deal and are at the stage of personal terms. Hopefully, we might be able to get him registered in time to play in the cup,” Allardyce said.
The 26-year-old, who netted 41 times in 96 appearances for Besiktas, will become Allardyce’s first signing since he replaced Ronald Koeman in November.