"If by chance in 2018 the World Cup were to go to Europe, I'm quite sure, with the English luck as it is, they won't get it," Warner said.
"It'll be Italy, Spain, or it might even be France. Nobody in Europe likes England. England, who invented the sport, has never had any impact on world football."
Thompson, who is also a FIFA vice president, said he had written
to Warner to ask for an explanation of his "potentially explosive remarks."
FA "first class"
England's 2018 World Cup ambassador Richard Caborn also hit back at Warner.
"What is unfortunate is that Jack has attacked England and particularly the FA - and the FA has done a first-class job," he told the BBC World Service.
"They have got 30 schemes running around the world, education programmes, on six continents.
"Indeed in Trinidad, for the last seven years they have had three workshops running which have been incredibly well-received by Concacaf, so it's not quite true what Jack is saying."
FA chief executive Brian Barwick was also "disappointed" by Warner's remarks.
"On Uefa, we are, alongside Germany, the most well-represented country in football across committees and working groups.
"I think frankly they are personal remarks," he added.
"I think Fifa will distance themselves from them very quickly."