The Guardian reported that Everton have beat off competition from several Premier League sides to agree a deal for Besiktas striker Tosun.
And they are now looking to January for reinforcements, with Tosun, who has 14 goals this season, chief among the targets.
"We are interested in him [Cenk Tosun] and many others", Allardyce said during his press conference ahead of Everton's trip to 18th-place Bournemouth", Allardyce told reporters.
Everton have struggled for goals this season and it has been a contributing factor to their poor form in the campaign before Allardyce's arrival in November. Hopefully, in the future, we can send more players to the top clubs around the world.
"We have also started to struggle to create chances and score goals in recent matches".
"Young Dominic (Calvert-Lewin) is battling away but his goal record is still four in the Premier League, with a few assists".
"We need to strengthen in our goalscoring areas", Allardyce said.
"It is still one of the major problems that we have".
"Yes I would like him but with the amount of midfield players I have at the moment I'd have to move some player out to bring a player like Steven in", he added. In actual fact, at the end of the game, they were lucky to get a draw - it was quite a freaky game in the conditions.
"But, he is training with us on a regular basis and he's very close to being available for selection". "They have been out far too long, a lot of the players". They've been able to get a couple of key results.
Full-back duo Leighton Baines and Seamus Coleman will remain out for the foreseeable with neither progressing how the Blues boss would like.