They claim that the Blues have re-opened talks with the former England boss about taking over team affairs on a more permanent basis than David Unsworth after a run of just one win in seven games under the caretaker boss.
Allardyce was said to have held talks with the Everton hierarchy a few weeks ago but any negotiations stalled and the 63-year-old suggested that he didn't feel wanted at Goodison Park.
Despite a victory over Watford and a somewhat fortuitous draw at Crystal Palace the team's fortunes have worsened with a 5-1 defeat to Atalanta being followed by yesterday's 4-1 drubbing by Southampton, perhaps forcing the Blues' Board to revisit what was, perhaps, merely a contingency in the early days following Koeman's dismissal.
Allardyce is reportedly flying back from a trip to watch the Formula One Grand Prix in Dubai and could hold face-to-face talks with Everton tomorrow.
The former Bolton, Newcastle, West Ham and Crystal Palace boss is said by Sky to be on a shortlist of three names which also includes Republic of Ireland head coach, O'Neill, and Shakhtar Donetsk manager, Fonseca.
It is unclear whether Fonseca has been interviewed by the club but Sky Sports claim discussions were held with his agent on Friday.
O'Neill, meanwhile, had the shortest odds with the bookmakers prior to these latest reports surrounding Allardyce.