Daily Telegraph has a footage from a meeting in August between Allardyce and two undercover reporters purporting to be from a Far East firm.
Sam Allardyce has been sacked as England manager after 67 days in the job, Sky Sports reported on Tuesday.
Allardyce, 61, was caught up in a newspaper sting in which he is alleged to have met a bogus consortium of Far East businessmen seeking advice on the Premier League's billion-pound transfer market.
During meetings with undercover reporters from The Daily Telegraph, Allardyce is reported to have criticised his predecessor Roy Hodgson, referring to him as "Woy" and saying he "hasn't got the personality" for public speaking.
Former Bolton Wanderers, West Ham United and Sunderland boss Allardyce took charge after Hodgson resigned in June following England's 2-1 loss to Iceland, and has managed only one competitive game to date - a 1-0 World Cup qualifying victory over Slovakia on September 4.
The Daily Telegraph has footage claiming to be from a meeting in August between Allardyce and two undercover reporters purporting to be from a Far East firm.
In the video, Allardyce explains to the two men how they might "get round" rules regarding third-party ownership of players, which was banned by FIFA last year and has been outlawed in England's top flight since 2008.
Allardyce used the example of Enner Valencia, a player he claims was owned by a third-party when he brought him to West Ham from Mexican club Pachuca in 2014.
"It's not a problem... we got Valencia in. He was third-party owned when we bought him from Mexico," Allardyce said, before acknowledging that his registration had to be purchased outright before moving to the Premier League.
The newspaper alleges that Allardyce reached an agreement worth £400,000 to represent the fictitious firm to Far East investors and be a keynote speaker at certain events, though he makes it clear than any arrangement would need FA clearance.
The 61-year-old is then claimed to have critised predecessor Roy Hodgson and his assistant Gary Neville for England's Euro 2016 exit to Iceland, while he brands the FA's expensive rebuilding of Wembley as "stupid".
Allardyce has yet to respond to the allegations, while FA chairman Greg Clarke says he wants all the information surrounding the issue to be assessed. An FA meeting is understood to be taking place on Tuesday.
"I want all the facts, to hear everything from everyone and make a judgement about what to do," Clarke told The Daily Mail.
"Natural justice requires us to get to the bottom of the issues before we make any decision. It is not appropriate to pre-judge the issue. With things like this you have to take a deep breath."