Allardyce denied he had taken part in talks with the club around two weeks ago and reportedly cited the club’s lack of urgency to secure a deal with a new manager to be his reason for pulling out of the race.
With the Blues looking set for a relegation battle at the bottom end of the Premier League table, they have turned to the 63-year-old who is a specialist in keeping teams away from relegation.
If appointed manager, Allardyce is unlikely to ever go down as a fans favourite at Goodison Park even if he does keep the club from sliding out of the Premier League.
After allowing Koeman and Director of Football Steve Walsh to spend £150 million over the summer, Moshiri and Everton fans had hopes of pushing on this season - building on the return to Europa League finish that the Dutch boss provided last campaign.
Instead, Everton sit 16th in the table, two points above the relegation zone with three wins in the league all season.
They also sit as the team to concede the most goals so far this season, conceding 28 in 13 games, with a goal difference of minus 15.
To stop the bleeding, it seems as if the Blues will go back to basics under Allardyce who is known as a defensive boss who looks to maximise clean sheets before worrying about scoring at the other end.
Sky Sports have already suggested that former Leicester City manager Craig Shakespeare would join Allardyce at Goodison Park as his assistant.
That had been reports that Allardyce would consider Michael Appleton as his assistant but is likely to choose a different former Leicester coach.
The 63-year-old is a big believer in sports science and analytics and would likely bring in a team of his own for that side of the role.