The Sunderland manager is reportedly the only Englishman under consideration.
The Three Lions are currently looking for their next manager after Roy Hodgson resigned from the position last Monday.
He left the role after England were knocked out of Euro 2016 at the last 16 stage by minnows Iceland.
A number of names are likely to be considered by the FA as they seek to find the perfect successor to Hodgson.
There are the usual calls for the next appointment to be an English one, but according to the Independent Allardyce is the only Englishman in the running.
As far they are concerned, he is regarded as possessing the experience, track record and personality to take on the important job.
However, the Sunderland manager remains an outsider to land the job as the FA looks into appointing a foreign coach instead.
Arsenal manager Arsene Wenger is under consideration, as is compatriot Laurent Blanc, while USA manager Jurgen Klinsmann is currently the bookmakers favourite for the job.
It remains to be seen who England will approach, but Sunderland will certainly be hoping it is not Allardyce given his considerable success last season.