The Clarets made the shock decision to sack Sean Dyche and his entire backroom staff on Friday morning, with captain Ben Mee set to take charge of the club for their game against West Ham on Sunday.
Burnley want to bring in Allardyce to chart a great escape in the last eight games of the Premier League, but are reportedly facing a tough task in convincing him to return to management.
Keys conceded that eight games might be too little time for Allardyce to make a significant impact, but feels the incentive to keep Burnley up at the expense of Everton could entice him back.
The veteran broadcaster took to Twitter and wrote: “There’s probably too few games left for Sam to take it – but what a double whammy incentive he’d have – keep Burnley up and send Everton down.
“It would make for a fascinating finish.”
Allardyce took charge of Everton in November 2017 after they sacked Ronald Koeman following a disastrous start to the season.
He made them defensively solid and guided Everton to an eighth-place finish in the Premier League.
However, Everton chose to part ways with Allardyce as fans were dissatisfied with the team’s style of play.
