Reporters from The Sunday Times posed as lobbyists for a consortium of American companies who wanted to bring the tournament to the United States.
They approached Amos Adamu, a Nigerian who serves as a Fifa executive committee member.
He allegedly told reporters he wanted cash to build pitches in Nigeria.
According to The Sunday Times, Adamu who is president of the West African Football Union is said to have told the undercover journalists that he wanted $USD800,000 to build four artificial football pitches.
This would be completely against Fifa’s rules.
The footage shows Mr Adamu wanting money to be paid to him directly for endorsing the US bid.
In the video, he was asked whether the money for a “private project” would have an effect on the way he voted.
He replied: “Obviously, it will have an effect. Of course it will. Because certainly if you are to invest in that, that means you also want the vote.”
Reynald Temarii, president of the Oceania Football Confederation, is also alleged to have asked for a payment, in his case to finance a sports academy.
A statement from Fifa read: “Fifa and the Fifa Ethics Committee have closely monitored the bidding process for the 2018 and 2022 Fifa World Cups and will continue to do so.
“Fifa has already requested to receive all of the information and documents related to this matter and is awaiting to receive this material.
“In any case Fifa will immediately analyse the material available and only once this analysis has concluded will Fifa be able to decide on any potential next steps.
“In the meantime Fifa is not in a position to provide any further comments on this matter.”
A European nation will definitely host the 2018 World Cup after the United States – the last remaining non-European bidder pulled out of the race on Friday. Australia withdrew its candidature in June. Both will refocus their efforts on 2022.
England will now battle it out with bids from Russia, Belgium/Netherlands and Spain/Portugal.
A 24-strong committee will decide by secret ballot on December 2 who should host the tournaments.