The Mail reports that Allardyce was sounded out by Moshiri, and the two met for talks after the businessman liked what he heard regarding the former England boss' interest in the job, and there have also been talks with director of football Steve Walsh regarding the backroom team that Allardyce would want - which includes former Leicester City boss Craig Shakespeare.
The Mail adds however, that nothing has been decided upon - but the signs certainly suggest that Allardyce is in the driving seat to land the job just a few months after he left Crystal Palace after guiding the Eagles to safety in the Premier League.
Additionally, current Toffees caretaker David Unsworth is set for talks with the Everton hierarchy regarding his own future. The Mail reports that Unsworth has impressed the squad with the job that he has done so far - despite only overseeing one win in his first four games.
Despite impressing the players however, it appears that Unsworth may not be the man to take the job on a full-time basis, as the report certainly suggests that there is a desire to seriously consider Allardyce.
The 63-year-old is vastly experienced in the Premier League and knows what it takes to inspire drastic turnarounds in form, so it is not surprising that the Everton hierarchy appear to view Allardyce as a potentially ideal option given their current position in the table.