Allardyce was West Ham boss when the midfielder moved to Cardiff in September 2014 and the Welsh club launched an investigation after questions were raised over fees connected with the deal.
The Daily Telegraph report an agent is understood to have told the club the payment had been agreed with Allardyce.
Cardiff brought in an external company to investigate but did not find any evidence of the payment being made to Allardyce.
However, Cardiff failed to report the issue to the FA, which is in breach of the governing body's rules. The FA could now launch an investigation into Cardiff, which could lead to Allardyce being quizzed.