The two Premier League managers expect the World Cup to be moved from its regular summer period to later in the year in order to control safety fears over the weather conditions
Fifa's executive committee continued its two-day meeting in Zurich on Friday, discussing alleged poor treatment of migrant workers in the Middle Eastern nation and the soaring temperatures during the summer.
But with the date of the competition increasingly under scrutiny amid fears of fan and player safety, Manchester United manager Moyes foresees compromises.
"I think that there will be no choice that [the date of the World Cup] will have to move," he told reporters.
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"There will be lots of different things: add-ons, maybe you'll need to get rid of replays in the FA Cup, you'll probably need to maybe limit a Capital One Cup competition, maybe, something along those lines."
Similarly, the Arsenal manager backed a move to see the competition played in winter but believes that it needs to start at the end of January so that the December schedule is not affected.
"To me it looks reasonable to play in winter because the only thing that matters in these is the safety of people who go there and attend – the fans," Wenger explained to the press.
"The players can cope with the heat, they are prepared and they will be in great conditions but the fans? I'm not so sure. If it's played in winter you cannot sacrifice the Christmas period here.
"That means the World Cup – initially is planned to be played on January 20 onwards – will need to be pushed back because you release your players four weeks before the competition starts.
"If you want to keep the Christmas period alive, that is to then happen after January 1, four weeks later. That means the World Cup starts [at the] end of January [at the] earliest."