During meetings with undercover reporters from The Daily Telegraph, Allardyce is reported to have criticised his predecessor Roy Hodgson, calling him "Woy" and saying he "hasn’t got the personality" for public speaking.
Former Bolton Wanderers, West Ham United and Sunderland boss Allardyce replaced Hodgson as England manager in July.
Allardyce is also alleged to have said England’s players were underperforming because they had a "psychological barrier" and "can't cope", suggested players who were not being played for their club should not be picked for England and described the FA decision to redevelop Wembley Stadium as "stupid".
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Reuters was unable to independently verify the report.
An FA spokesman said: "We have asked The Daily Telegraph to provide us with the full facts in relation to this matter.”
(Writing by Ken Ferris; Editing by Mitch Phillips)
