The Toffees brought in the 26-year-old from Besiktas a week ago to bolster their faltering attack, parting with around £27m to get a deal over the line.
Having seen a number of players stutter after moving to a new country in the past, however, Everton boss Allardyce concedes that throwing Turkey international Tosun straight into the mix is a gamble of sorts.
"It is not easy at all, it might not work," he told reporters. "It is a massive problem in terms of you are hoping rather than being guaranteed he hits the ground running.
"When you look at the number of players who have been brought into this country from abroad – even in the summer – and not worked, particularly up front, it is a risk."
Tosun, who netted 14 goals in the first half of the campaign for Besiktas, is in line to make his Premier League debut against Tottenham Hotspur on Saturday evening.