An announcement on Allardyce's appointment is a matter of hours away.
Sky Sports News understands 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.
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.
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.
However, Allardyce did watch his side dismantle West Ham 4-0 at Goodison Park on Wednesday night, courtesy of a Wayne Rooney hat-trick, so there is reason to be optimistic. It is a massive six-pointer for both sides.
Everton sit 17th in the Premier League, just two points out of the relegation zone, and have already been eliminated from contention for the Europa League knockout rounds after claiming just one point from five matches.
The Toffees were expected to perform well this term thanks to the amount of spending the club made during the offseason.
The former Newcastle United, Bolton Wanderers and England manager has famously never been relegated from the Premier League, having taken over mid-season to keep Sunderland and Palace in the top division the past two campaigns.