The former England manager, who left Crystal Palace after saving the club from relegation at the end of last season, has also been on the hit list of West Brom, but the 63-year-old has distanced himself from the job.
Lee is a supporter of Everton's bitter local rivals Liverpool and, as well as being a player for the Reds, he was once a part of the coaching staff at Anfield, arriving there in 2008 to serve as assistant manager to Rafa Benitez. But the chance to turn around Everton's faltering ship has lured him back, and Allardyce is expected to have a watching brief at Goodison Park on Wednesday night.
Allardyce re-emerged as the leading contender for the job this week after Everton were forced to accept defeat in their attempts to persuade Watford to part with Marco Silva.More news: Israel to open its first embassy in Rwanda
Allardyce will take charge of the team in the next 48 hours.
Sam Allardyce will be at USM Finch Farm this afternoon to finalise terms on becoming the new manager of Everton Football Club.
With Unsworth overseeing his final match at home to West Ham tonight, Allardyce's first game in charge will be against Huddersfield Town at Goodison Park this Saturday.
Everton's chief problem is clear to see. But, after taking four points from their first two games, Everton managed the same tally from their next seven fixtures and Koeman was sacked.More news: Blake Griffin avoids season-ending knee injury
The Toffees hierarchy will feel they have gone for a safe pair of hands, with Allardyce having taken over both Palace and Sunderland in graver situations and led them to safety.
It's five weeks since the Toffees dismissed Dutchman Ronald Koeman thanks to a poor run of form that saw them slump to third from bottom in the Premier League.
Further forward, Allardyce will need to get Everton scoring more, something the team has struggled to do since selling Romelu Lukaku to Manchester United for 75 million pounds ($97 million) during the summer.More news: Dancing ballerinas help kick off Christmas at White House