Sharing a ground would make economical sense for both clubs, especially for Everton who do not have the same level of financial backing as their neighbours.
Ground sharing has been proven to work abroad, with both AC Milan and Inter Milan sharing the San Siro but it is understood to be an unpopular move with fans on Mersyside.
Both clubs have been considering a new stadium for a long time, but Liverpool seems to have stalled on a decision whether to rebuild Anfield or a new site on Stanley Park.
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Moyes told reporters: "I'd take a shared stadium, because I think Everton need it probably more than Liverpool - although it's not in our culture to share stadiums.
"But I have to say it does make sense. You could redevelop the whole area around Stanley Park, so for us it would definitely be worth it for both clubs.
"Everton, being honest, would say probably we would take it, but Liverpool not so, and that's probably where it will end up."
Any decision is likely to be a long way off, however, and Moyes was keen to talk about the present situation at Everton.
It appears that the Scot is happy to continue to at Goodison Park, although he did admit he would love to work with a big transfer kitty - something that current chairman Bill Kenwright cannot provide.
He added: "I'm Everton manager and I am planning for next season as that.
"I'm out at the Euros talking to my players, looking at others, and I and the chairman will be trying to keep things rolling along.
"Bill has not hidden the fact he is trying to sell the club and get money in to invest. We just have to keep going, keep looking for players, like [Nikica] Jelavic at a decent price.
"At Everton, I think we could make a difference without too much money, we don't necessarily need £50m-60m... but if someone did give me a wad of cash I'd be delighted!"