Allardyce's most recent managerial position was at Crystal Palace - he resigned in May
Sam Allardyce is back under consideration for the Everton job.
The club's search for a new manager has taken on fresh urgency after Sunday's 4-1 defeat at Southampton.
Allardyce was an early candidate to succeed Ronald Koeman, who was sacked on 23 October, but publicly withdrew after Everton delayed making an offer.
Everton, whose pursuit of Watford's Marco Silva has reached a dead end, have lost five out of seven games since Koeman left.
Allardyce, 63, has been out of football since leaving Crystal Palace at the end of last season.
He saved Palace from relegation, having left the England job in September 2016 after just one match in charge.
Dutchman Koeman was sacked by Everton after the club slipped to 18th in the Premier League following a 5-2 home defeat by Arsenal.
David Unsworth - the club's Under-23s manager, who had two spells at Goodison Park as a defender - has been in caretaker-charge since then.
He has lifted Everton to 16th, collecting four points from his four league matches.
Analysis
Everton's drawn-out hunt for a manager to take over from Ronald Koeman has left caretaker-manager David Unsworth in charge for seven matches, from which he has gained just one win.
The need for major shareholder Farhad Moshiri and chairman Bill Kenwright to appoint a manager has become more desperate after a 5-1 humiliation at home to Atalanta in the Europa League on Thursday followed by the 4-1 thrashing at Southampton on Sunday.
Everton spent more than £140m in the summer but failed to replace top scorer Romelu Lukaku after his move to Manchester United. They are two points above the relegation places - and third-from-bottom West Ham United, who they face at Goodison Park on Wednesday.
Source: bbc.com