Unsworth is the current caretaker boss at Goodison Park, stepping up from his job as under-23s coach when Ronald Koeman was sacked last month.
The Mirror reports however, that Allardyce is in contention for the post. And not only may Unsworth end up not getting the first-team job on a permanent basis, he may find himself out of work altogether.
According to the Mirror, despite former owner Bill Kenwright's desire to see the 44-year-old stay at the club, Allardyce's plans to bring Craig Shakespeare with him and form his own backroom team may mean that there is no space for Unsworth or Royle - who has also stepped into the breach following Koeman's departure.
It is not made clear by the Mirror whether Unsworth will have the chance to revert to his old role with the academy side if Big Sam does end up taking the job, but perhaps the Toffees caretaker may not be keen anyway after having a taste of first-team management in recent weeks.
Unsworth has overseen one win and three defeats during his four games in charge so far, but the dramatic events of the 3-2 win over Watford in their most recent game before the international break may have left him feeling extremely optimistic that he could have the potential to succeed as a first-team manager elsewhere if the Everton job is not available to him.