So could the former England manager finally end a long-running transfer saga in January?
The rumour mill might have slowed in the past 12 months, but Sunderland are still feeling the effects of Everton's failed bid to sign Lamine Kone - per Sky Sports - in the weeks following Allardyce's departure from the Stadium of Light.
Kone had starred under Allardyce, turning in a string of inspired performances to drag the Black Cats to Premier League safety.
But when the manager moved on, Sunderland's star defender also sought the exit door - only for the club to firmly stand in his way.
It did not end well for Sunderland or Kone, who are both now stranded in the Championship after a dismal campaign under David Moyes.
Allardyce is now back in the game, though, and the Black Cats could look to cash in on a saleable asset, so could Kone finally sign for Everton?
The Toffees could certainly use another centre back option, with Phil Jagielka and Ashley Williams ageing not all too gracefully.
Allardyce has signed players he knows well throughout his managerial career - think Abdoulaye Faye, Kevin Nolan and Patrick van Aanholt - and so this move could well work for all parties.