Lee, 58, is a former Liverpool midfielder who played for the Reds in the 1970s and the 1980s.
The former England international won the old Football League First Division title thrice, the League Cup on four occasions and the European Cup once with Liverpool.
Former Liverpool defender Carragher has given his verdict on Allardyce and Lee on Twitter, suggesting that they would be good appointments for Everton.
Everton will face West Ham United at Goodison Park in the Premier League this evening.
It will be David Moyes’s second game in charge of West Ham and will see him return to his former club Everton.
Ahead of the game, the Scotsman has claimed that in his last season at Everton, he was one striker short of finishing in the Champions League places.
“In my last year we were desperate for a striker,” Moyes told The Daily Mail. “Everton were a striker short of being top four. You were still needing somewhere like £10 or £15m to get a top striker. We had a million left.”