Manager Sam Allardyce confirmed Thursday that the Premier League club have agreed a fee with Besiktas -- reportedly $36.6 million (£27 million) -- for the 26-year-old.
Allardyce even suggested the forward could be registered in time for the Merseyside Derby tie against Liverpool in the FA Cup on Friday.
"It is just personal terms now, the final stage," he told a press conference. "After long negotiations we are nearly there.
"He might even be registered before the FA Cup but securing the player is the most important thing because a lack of goals has been our biggest area of concern, and before I got here.
"With his pedigree and his age -- we've not got many players in that elite age group in our squad -- hopefully he will show he is good value for money."
Tosun -- who has earned 25 caps for Turkey -- began his career at Bundesliga club Eintracht Frankfurt, where he made just one appearance.
After two years of trying to break into the first team, the forward made the move to Turkey and joined Gaziantepspor in 2011, enjoying four successful seasons and three times reaching double figures.
His performances convinced Turkish giants Besiktas to sign him on a five-year deal in 2014 and he went on to score 24 goals in a Championship-winning campaign.
Tosun has continued his prolific form in this season's Champions League, netting four times in six matches to lead Beskitas to the top of Group G and set a points record for a Turkish team in the competition.
He is now the most expensive export in Besiktas' history and adds much needed firepower to an Everton side that has scored just 25 goals in 22 Premier League games this season.
Everton have so far failed to replace Romelu Lukaku who moved to Manchester United in the summer, spending the majority of $101 million transfer fee on midfielder Gylfi Sigurdsson.
"If we get the transfer done and he comes and scores goals, it will be exceptional business," Allardyce said.
"The club has searched high and low for a striker for a long time, since before I got here, and he was the one we focused on. He wants to come here."
Tosun also becomes Allardyce's first signing since he was appointed Everton manager at the end of November.
The former England manager joined with the club languishing in the bottom half of the table, but led the team on a seven-game unbeaten run at the start of his tenure.