This article suggests May which would absolutely be do-able. They'd just have to start all of the major leagues a month earlier in the previous year so that everything would end in April (including the Champions League Final).
The issue is that having it in May doesn't solve the heat problem so I don't understand why there is speculation around May. The average high is still 100 degrees as opposed to 106 in June/July. The November suggestion makes more sense climate-wise (a "cool" 84 degrees) but that would ruin league play.
Another issue with a May tournament would be a shortened preparation (leagues would probably do everything they can to be the nearest to usual schedule) just like in 2002. And as much as underdog stories are enjoyables, it's still a huge loss for football to not see some of the best teams at their maximum.
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