Allardyce is one of the frontrunners to get the job full time after Ronald Koeman was sacked.
Former England boss Allardyce is said to have spoken to Everton majority shareholder Farhad Moshiri about the vacancy, claim Sky Sports.
The revelation comes as ex-Everton manager David Moyes flies to London to take up the West Ham job.
Everton have been seeking a manager since sacking Dutchman Koeman on October 23.
Burnley boss Sean Dyche is another option for the Toffees, although the Clarets would likely demand a hefty compensation fee to let him go.
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Discussing Everton on Saturday, Allardyce said: “Everton are obviously in a bad place at the moment and, like every football club that is struggling, it’s mainly down to a loss of belief.
“The players’ loss of belief in themselves and their lack of confidence means that whoever becomes the manager of Everton, the first thing they need to do is re-instil that confidence and morale in them again.
“When a player plays with confidence, that’s when he’s playing to the best of his ability. At the minute, I can’t see that with that set of Everton players. It’s completely gone.
“At any football club, that should be your top goal and your main focus before anything else, keeping your team mentally happy.”
Allardyce has also been linked with the Hammers post in recent weeks, with pressure slowly mounting on Bilic until today’s sacking.
Both Everton and West Ham have pulled the trigger on their managers in a desperate bid to recover their seasons.
Express Sport understands Moyes will not be heading back to Goodison and instead will sign a short-term contract at the London Stadium.
Allardyce, like Moyes, has form in keeping sides in the top flight.
He oversaw Bolton’s rise to the Premier League and also saved West Ham from the drop, before being let go in 2015.
He was appointed England manager after a successful stint at Sunderland saw the Black Cats steer clear from danger.