Sky Sports News on Monday that the Toffees had reopened discussions with the 63-year-old about succeeding Ronald Koeman, after original dialogue broke down some two weeks ago.
An agreement could not be reached at the time but, after Everton failed with an attempt for Watford boss Marco Silva, progress was made over the key aspect of the length of a contract for Allardyce earlier this week.
It is also understood that Shakhtar Donetsk manager Paulo Fonseca was also interviewed for the role, while Republic of Ireland's Martin O'Neill was also being considered, but Everton owner Farhad Moshiri was expected to make a firm decision on Tuesday.
Allardyce, who has been the preferred choice of director of football Steve Walsh, is close to managing his seventh Premier League club after leaving his role as boss of Crystal Palace in the summer.
He is likely to be joined by former Leicester boss Craig Shakespeare, who was his right-hand man during the ill-fated four-month spell with England.
Everton find themselves in a relegation battle after a torrid spell of five defeats in seven games under caretaker boss David Unsworth, and Allardyce is set to arrive having kept Palace up against the odds last term.
Palace chairman Steve Parish agreed to his departure and it is believed the south London club would receive a compensation package of around £2m if Allardyce takes a new job before next summer.
Allardyce has been on holiday in Dubai, having enjoyed a trip to the Abu Dhabi Grand Prix over the weekend.
Sky Sports