Everton shelled out a reported £27m for the Besiktas forward in the winter transfer window, the third highest total they have ever paid for a player.
That fee, combined with an expected difficult adaptation period to English football, meant that pressure was already going to be heaped on the 26-year-old's shoulders.
Since then, Allardyce has twice left the Turk on the bench behind unanimous flop Oumar Niasse.
Rather embarrassingly, following Everton's 1-0 home loss to Watford, Big Sam went on to tell the Liverpool Echo: "We didn't have a finisher on the pitch. Watford did."
That quote now joins a collection of comical comments, including his claim that 'Ademola Lookman would not be allowed to go out on loan' - just two days before he made the switch to RB Leipzig.
Tosun arrived with a record greater than a goal every two league games for Turkish champions Besiktas.
While his displays have shown signs of encouragement, a return of one goal during a spell of three losses in four games does not yet match the praise.
With Allardyce under scrutiny from the majority of Toffees fans, it is likely that plenty will cite this statement as part of the reason he must go in the near future.