British media reports say that the Premier League club will pay a fee of around 27 million pounds for the prolific Turkey international.
"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 told a news conference on Thursday.
"There is a medical and all sorts to go through so that might be very tight. Securing the player is the most important thing obviously because it has been the club's biggest area of concern."
Everton are ninth in the league after recovering from a poor start to the season but have been toothless in attack, scoring just 25 goals in 22 league games.
Allardyce also said that the Merseyside club would not send midfielder Yannick Bolasie out on loan in the January transfer window while forward Kevin Mirallas could be allowed to leave.
The 63-year-old boss confirmed that England international Ross Barkley was likely to leave Goodison Park this month and said Everton were yet to receive any bids for struggling striker Sandro Ramirez.
Defender Michael Keane is the only doubt for Everton as Allardyce's side hope to end a winless run against Liverpool, whom they last beat in a league match in 2010, and progress to the fourth round of the FA Cup.
"We'll go for it," Allardyce added.
"It's win or bust, simple as that. It's a glory game and you try and win it. A draw is not good enough."
Everton drew 1-1 with Liverpool at Anfield in last month's league fixture.
(Reporting by Aditi Prakash in Bengaluru, editing by Pritha Sarkar)
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