Allardyce has been given a second chance to manage the Toffees, after he initially rejected them.
The former England boss turned the Toffees down earlier this month, after deciding that Everton had not shown him that he was a priority.
However, with Everton’s form taking a turn for the worse recently, the Merseyside club have returned to make an offer for Allardyce.
And the 63-year-old is said to be unlikely to turn town down Everton for a second time.
Everton have turned to Allardyce, as they are fearful that they could get dragged into a relegation fight.
The Toffees are in 16th place in the Premier League table, and lost their last match 4-1 against Southampton.
Everton have spent six weeks searching for a replacement for Ronald Koeman, and the uncertainty over the managerial situation seems to have had a negative impact on the players.
Everton are next in action tomorrow evening, when they take on Leicester City.