The striker travelled to England on Saturday and had been expected to complete his move to Goodison Park tomorrow after a deal worth a reported £25m had been struck with the Turkish club.
It appears, however, as though Besiktas have upped their asking price after being told to increase their bid for the man they want to replace Tosun.
"We've done all we can to try to secure the player," Allardyce said after today's defeat to Manchester United during which his team failed to register a shot on target for the second home game in succession. "Whether that is enough, only time will tell in the next 24 hours.
"Our [bid] is in there and it's up to them whether they accept our deal or not.
"Our ratio of chances created is very limited and that's because we need to strengthen that front line. I think we all know that.
"We're trying to do the hardest job of all in trying to secure a player in that position... it may not get achieved this January."