The Three Lions are currently looking for their next manager following Roy Hodgson’s departure last month.
He left following the three Lions' dismal Euro 2016 exit at the hands of Iceland, arguably the country’s worst defeat for a number of years.
A series of managers have subsequently been linked to the role, with the Independent suggesting that Allardyce is the only Englishman that will be considered by the FA.
And according to the Mirror, Sunderland staff are convinced that he would depart should the job be officially offered to him.
According to the report, they fear that the growing frustration from the manager in regards to Sunderland's failure to move in the transfer window could see him seek an exit.
The Black Cats are yet to make a signing this summer, with last Friday seeing a deal for defender Davide Santon fall through over a disagreement with Inter Milan.
Their lack of movement has allegedly frustrated Allardyce and staff fear that could lead to him taking the England job should it be offered.