President Sepp Blatter admitted the scandal has had a "very negative impact" on the world governing body.
Reporters from The Sunday Times posed as lobbyists for a consortium of American companies who wanted to bring the tournament to the United States.
Amos Adamu and Reynald Temarii allegedly asked for payments.
Adamu, who serves as a Fifa executive committee member, allegedly said he wanted $800,000 (£500,000) to build four artificial football pitches.This would be completely against Fifa's rules.
Adamu, who is president of the West African Football Union, 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."

The Obamas were to make campaign on Sunday for Gov. Ted Strickland and raise money for the Democratic National Committee.Their day was set to end in Columbus at a large, nighttime rally on the campus of Ohio State University aimed at firing up the young Democrats and first-time voters who helped Obama carry Ohio on his way to the White House in 2008.
Obama has been campaigning coast to coast as the Nov. 2 midterm elections fast approach, trying to convince wary voters that his policies have put the nation's economy on a path toward recovery. But even the president has acknowledged that people are angry and frustrated in the face of 9.6 percent unemployment.
"There is no doubt that this a difficult election. That's because we've been through an incredibly difficult time as a nation," Obama said Saturday at a campaign rally in Boston for Gov. Deval Patrick.
Though the first lady campaigned heavily for her husband during his presidential bid, she's largely stayed out of politics since moving to the White House. Her popularity has stayed high, while the president's has fallen, making her a valuable asset on the campaign trail for Democrats.
Mrs. Obama's campaign message has echoed her husband's — saying he can't make good on the promises he made during the presidential campaign unless voters keep Democrats in control of the House and Senate.
She has urged supporters to summon the same enthusiasm that helped sweep her husband and many congressional Democrats into office last election cycle.
"There's an excitement about what this president is trying to do. There's an energy around it," his spokesman, Robert Gibbs, said Sunday on NBC's "Meet the Press."
Obama said Saturday he still believes Democrats can retain control Congress, though recent polls suggest Republicans may well retake the House and make major gains in the Senate.
Gibbs said Democratic candidates "have done a remarkably good job in a tough political environment, and I think that come election night, we'll retain control of both the House and the Senate."
For Obama, it's the start of a busy campaign period for the president, with a mix of fundraises and rallies.
Fresh from three straight days of events in Delaware, Massachusetts and Ohio, Obama heads to the West for stops in Portland, Ore., on Wednesday, Seattle and San Francisco on Thursday, Los Angeles and Las Vegas on Friday and Minneapolis next Saturday.
Obama probably will spent Election Day in the nation's capital, and has requested an absentee ballot to vote in his home state of Illinois.