Everton sent a delegation to Turkey on Friday, led by director of football Steve Walsh, to try to force through the deal amid interest from elsewhere in Europe.
Allardyce admits that a deal for Tosun, 26, is "in the wind", adding: "The interest from other clubs around Europe is our biggest worry if we are to secure that transfer."
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.
"I would take Steven N'Zonzi tomorrow," he said. "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."