Besiktas fans appeared to wave goodbye to the 26-year-old forward after he netted for them against Osmanlispor in a Turkish Cup tie on Thursday night.
"We are interested in him and many others", Allardyce said during his press conference ahead of Everton's trip to 18-place Bournemouth.
"We would be actively looking for a frontman in this window because I need more goals in the side than we have at the moment".
New year, blue Moon
According to NASA , "A supermoon is a Moon that is full when it is also at or near its closest point in its orbit around Earth". Full moons are usually separated by 29 days, but the extra "blue moon" was able to squeeze in this January, which has 31 days.
"We are struggling to score goals".
"We've got more penalties than anybody else in the Premier League, which has contributed to a few points".
"Sandro hasn't picked up the pace of Premier League and hasn't been able to show what he could do in Spain, Oumar [Niasse] has scored the odd goal but has never been a regular and was loaned out to Hull City while Dominic Calvert-Lewin is battling away but his goal record is around four".
Despite making two appearances for Germany's Under-21 side, Tosun opted to play for Turkey at worldwide level and has eight goals from his 24 caps.
A striker and two full-backs remain the manager's January priorities, with a deal for Sevilla midfielder Steven N'Zonzi, who played for him at Blackburn, only likely if he can sell first to create room in his squad.