The world governing body was recently the subject of an anti-corruption investigation, particularly focused on the decision to award the 2022 World Cup to Qatar, with the outcome due to be delivered in September.
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"[England] are too straight in a way - and I am not saying that as a criticism," the ex-striker told GQ. "We should stand up for what is right and, if that means never hosting the World Cup again, fine.
"The only way it might change is for the clean countries to turn around and say: 'You know what, we are not taking part in your tournaments.' But I can’t see it happening. There is too much at stake."
Lineker was part of England’s bid team for the 2018 World Cup – which was awarded to Russia on the same day that Qatar was unveiled as the following host, but feels it was always going to be a futile campaign.
"I was with David Beckham having a burger the night before the Qatar decision," he explained.
"We were out trying to get support. Prince William and [prime minister] David Cameron were out there too, and I said to Becks: 'We are the only country doing this. The whole thing smells; it is a done deal.'
"It makes you feel sick, actually, the whole Fifa thing. The corruption at the top level is nauseating. Sepp Blatter likewise has run it like a dictatorship for so long and he comes out with so much nonsense."
Lineker, who scored 48 goals in 80 internationals for England, added that he is an advocate for the tournament being played all over the globe, but wants decisions made in a proper manner.
"[The Qatar choice was] ludicrous. I am not against it going to different parts of the world, but you would have thought they might have known it would be very hot in summer. You have to be careful what you say, but the corruption is just... yuk."