Undercover reporters for a national newspaper claimed to have caught Nigerian Amos Adamu, a Fifa executive committee member, on film as he said he would guarantee his vote in exchange for 500,000.
The reporters posed as lobbyists for a consortium of private American companies which wanted to help secure the World Cup for the US.
The US pulled out of the race on Friday, saying it wanted to focus on bidding for the 2022 tournament. The reporters involved in the story emphasised they had no links to the bid and that the US committee's campaign had been completely above board.
At an initial meeting in London with Adamu, also president of the West African Football Union, he is said to have told them he wanted money to build four artificial football pitches in his home country.
The newspaper said a second member of the committee, Reynald Temarii, who is president of the Oceania Football Confederation, asked to be paid for his vote as well. In his case, they said he wanted the money to finance a sports academy.
Fifa has asked for all material relating to the allegations.
A spokesman said: "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."
England is one of four bids still in the running for the 2018 World Cup alongside Russia and joint bids from Belgium-Holland and Spain-Portugal.
Foreign Secretary William Hague said the Government was "very disturbed" to read of the allegations. "Britain, whatever happens and whatever other countries do, will deal with this in a correct way doing absolutely nothing corrupt, paying no bribes and not corrupting this system in any way, " he said.