Allardyce was unwilling to offer the winger any assurances when questioned about his future at Friday's weekly press conference, insisting that any bids forthcoming will be dealt with by the club's chairman and director of football.
"I don't know [about his future]. I think I will probably find out at some stage or other but obviously that lies in the hands of the chairman and Steve Walsh," he told reporters. "They handle all that side of it, all those negotiations if and when they go on."
Lennon, who joined Everton from Tottenham Hotspur in 2015, has yet to play a direct part in a single goal in 18 league and cup outings this season.