Allardyce has been out of work since parting company with England last year, but speculation has begun to mount about where the 62-year-old boss could head next.
Allardyce was quick to dismiss taking over at Leicester City, suggesting that he didn’t feel it was the right time to take over at the King Power Stadium.
But when asked about whether he would be interested in taking over from Ronald Koeman at Everton he was far more coy.
He said: “I can’t speculate about that. I’m fully aware of the pressure. I’m not commenting on [Ronald] Koeman or Everton whatsoever.”
Koeman has come under serious pressure at Everton, with the Toffees struggling for form.
Everton invested heavily in their first-team squad over the summer, but were unable to bring in the requisite quality to help them improve.
Everton are currently in 16th place in the Premier League table, and Koeman has frustrated supporters with his tactical decisions throughout the campaign to date.
The Toffees are next in action on Thursday evening, when they take on Lyon in the Europa League.