Dyke says FIFA has two choices - to move the World Cup to a cooler part of the year or give the tournament to another country - but believes a winter World Cup is more likely.
It is the first time the FA has taken a formal position on the controversial issue, although Dyke's predecessor David Bernstein said in June that any plans to move the World Cup to the winter were "fundamentally flawed".
The latest statement could lead to some differences with the Premier League, whose chief executive Richard Scudamore strongly opposes any move to change the World Cup to the winter due to the disruption it will cause to domestic club competitions over three years.
Dyke said: "I don't know how many people have been to Qatar in June. I have. The one thing I can tell you is you can't play a football tournament in Qatar in June.
"Even if all the stadia are air-conditioned - which seems a bit strange in terms of the green policy - it would be impossible for the fans just to go out there, wander around in that sort of heat.
"The FA's position will be you can't play it in summer in Qatar. FIFA therefore has two choices; you either move it in time or to another location. I suspect either ends up in some sort of litigation but then someone should have worked that out in 2010 when it was awarded.
"It is then up to FIFA to decide whether to move it from Qatar or move it into autumn or winter. That's what the discussions will be over the next few months.
"I understand the reaction of the Premier League to not want to move it. I have some sympathy with them. They didn't have to choose to give it to Qatar in the summer but that's where it is. It will either have to be moved out of the summer or another location. I suspect the former is more likely than the latter."
In terms of the next World Cup, Dyke is optimistic England will be there in Brazil next year.
He does accept, however, that mistakes have been made in the approach to tournaments such as the European Under-21 Championship, where England performed poorly.
There is a feeling in the FA that some clubs were not supportive in terms of releasing players, and that a number of players who did join the squad did not take the tournament as seriously as they should have.
Dyke, who has spent an evening in the pub with England boss Roy Hodgson to discuss the various issues, added: "Roy lives up the road from me so we went and had an evening's drinking in Richmond, talking about the future, the importance of this year.
"There are four important competitive internationals and we are all pretty optimistic we are going to get to Brazil. Some people think it might be via the play-offs, I'm not so sure, I think we might get straight there. Roy is certainly confident. So are we at the FA that we will get to Brazil.
"We went into a couple of tournaments in the summer where we didn't do too well, particularly the Under-21s. We are now going through the process of appointing an Under-21 manager and the way it works.
"It needs to be more joined up, the FA and football structure, than it is now, right the way from young kids through. We need to make it very clear which tournaments we are going to take very seriously. In the summer I didn't think we made that clear and I think that was a mistake."