The Black Cats boss has received an approach to manage the England national team.
Allardyce this week held talks with the Football Association with a view to succeeding Roy Hodgson. And while the Sunderland hierarchy have reportedly been informed of the organisation's intention to interview other candidates, contingency plans are said to have been drawn up.
Former England manager Roy Hodgson resigned last month
And the Telegraph states it is looking increasingly likely that the former West Ham United and Bolton Wanderers manager will be the Three Lions' next boss.
As such, Sunderland will reportedly make tentative enquiries - through intermediaries - to gauge whether Moyes would be interested in succeeding the 61-year-old. And it's claimed the former Manchester United boss is expected to welcome an approach.
David Moyes pictured during his final game as Manchester United manager
Moyes has been out of work since being sacked by La Liga outfit Real Sociedad last November, but previously spent 11 years at Everton, where he gained the reputation that led to him being cherry picked to succeed Sir Alex Ferguson at Old Trafford by the man himself.
Former United assistant boss Ryan Giggs is also listed as a contender for the Stadium of Light hot seat, along with Burnley chief Sean Dyche.
Former Manchester United assistant manager Ryan Giggs could also be targeted