David Gold (right) says England should consider Sam Allardyce to be their new manager
Allardyce worked under West Ham co-owner Gold at the Upton Park club for four years.
The 61-year-old was let go at the end of last season, with Gold and fellow owner David Sullivan appointing Slaven Bilic as his replacement.
Allardyce took over at Sunderland last October and kept the Black Cats in the Premier League.
After Euro loss, Hodgson: 'I didn't see England defeat coming'
England boss Hodgson quit immediately quit after his side's shock 2-1 defeat to Iceland which saw them crash out out Euro 2016.
USA boss and former Germany international Jurgen Klinsmann is also a candidate for the role, as are Eddie Howe, Glenn Hoddle and Steve Bruce.
Allardyce is preparing his Sunderland side for the new season but is yet to make any fresh signings this summer.
Sunderland won nine of 30 games under the Englishman last season to avoid relegation by two points, with Aston Villa, Norwich and rivals Newcastle dropping into the Championship.
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