Allardyce, 63, has been out of football since leaving Crystal Palace at the end of last season.
Koeman was sacked by Everton on 23 October after the club slipped to 18th in the Premier League following a 5-2 home defeat by Arsenal.
It has now been over a month since the dismissal of Ronald Koeman, and having given caretaker manager David Unsworth a fair shot at the job, it is clear that the Merseyside club are in desperate need of a spark and huge boost of confidence.More news: Derrick Rose 'Evaluating His Future In Basketball' While Away From Cavaliers
However, a run of only one win in seven games under Unsworth, including a 5-1 home defeat to Atalanta and a 4-1 loss at Southampton on Sunday, prompted Everton to resume negotiations with Allardyce over the weekend.
Everton spent more than £140m in the summer but failed to replace top scorer Romelu Lukaku after his move to Manchester United.
The search for a replacement boss then descended into farce as Allardyce initially pulled out of talks because he wanted a deal beyond the end of this season, which major shareholder Farhad Moshiri and chairman Bill Kenwright did not want to offer.
They are two points above the relegation places and third-from-bottom West Ham United, who they face at Goodison Park on Wednesday.More news: Penn State Women's Volleyball Earns No. 1 Seed in NCAA Tournament
"When confidence is low, it takes a bit of time to build that back up".
"Maybe the next step is for "Unsy" is to try and get a decent job which would give him a chance of getting into management level and get a few years under his belt". He would utilise him to the best of his ability because he is someone he will respect.
They then turned their attention to Watford boss Marco Silva, the the Premier League club rejected their advances out of hand despite the offer of compensation.
West Ham are ready to make a move for Krasnodar striker Fyodor Smolov, and, at the same time, according to the Liverpool Echo, Everton have been linked with a move for Besiktas playmaker Oguzhan Ozyakup, who was once dubbed "the new Zinedine Zidane". It will be "you do what I tell you".More news: Sindhu Eases Into Hong Kong Open Final
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