Reports emerged yesterday that a group of key All Blacks players told New Zealand Rugby they don't want to leave their families for up to 10 weeks to play in the competition, which could include spending Christmas in quarantine.
Among the players named were first-fives Mo'unga and Beauden Barrett, halfback TJ Perenara and winger Sevu Reece.
Both Mo'unga and Perenara recently had newborn babies, while Barrett and Reece are also expecting their first children.
The reports came after Sanzaar announced that Australia will host the Rugby Championship in November and December, with the All Blacks due to play the Wallabies in two Bledisloe Cup tests in New Zealand in October.
While dates for the Rugby Championship haven't been confirmed, reports suggest All Blacks players could be forced to spend Christmas in quarantine due to New Zealand's 14-day quarantine rules.
Speaking to media today, Mo'unga said he was shocked by the reports.
"It was quite laughable really, it was the first I had heard of it," Mo'unga said.
"It's a tricky one because as players we are made accountable for the things we say and do and it's quite tricky when speculations come out like that and it can be harmful and sad for family members during these uncertain times.
"I'm sure players with newborn babies, soon-to-be-fathers will have conversations and make those decisions in their own time with [All Blacks coach] Ian Foster and the management."
Admitting it would be difficult to leave his young family behind, Mo'unga didn't express a preference for staying home.
"It definitely will be tough, you'd have to be a robot if that wasn't going to be tough for you and you're away from that certain amount of time but us footy players, we play rugby and we've got to do everything we can to put food on the table for our families as well."
Earlier today, Foster, too, denied the claims - although he did admit discussions were being held with players about the issue.
"I read that report yesterday and it was pretty frustrating that those headlines came out without any facts behind it," Foster told Newstalk ZB.
"Are we in trouble? No, we're not in trouble. Are we having conversations with players? Of course we are.
"We're trying to be responsible employers and talk to a whole lot of players. This is all new to everyone, taking players away for nine weeks where you can't get back and it's becoming increasingly obvious it's going to be hard to bring players over later.
"So it's a big chunk of time and we're just working it through with players."
Foster said he had spoken to the players named in the reports and insisted it wasn't true.
"I've had no player at all, and I've spoken to the players who were noted in the articles. I've spoken to all those players and I haven't heard anything like what has been reported. Right now, people just want to know what the facts are and we're gathering them up at the moment."
Meanwhile, Prime Minister Jacinda Ardern confirmed that the New Zealand Government will change quarantine rules to allow the Wallabies more time to prepare for the Bledisloe Cup in New Zealand.
Ardern said the director general of health, Dr Ashley Bloomfield, had advised that the Wallabies would now be able to start training after three days and as a full squad after six days, with previous protocols restricting the team to training in limited bubbles.
The Prime Minister also said she spoke to her Australian counterpart Scott Morrison last night to ensure the two Bledisloe Cup tests would go ahead, following claims from Wallabies coach Dave Rennie that the team may boycott the first test if they weren't allowed enough time to prepare.
Foster said he was "thrilled" with the change in quarantine rules to allow the Bledisloe tests to go ahead.
"Very thrilled with that even though we're bitterly disappointed we're losing the Rugby Championship but we're finished sulking about that," Foster said. "We're looking forward to it and now can't wait to sort out these Bleds. A little bit of water to go under the bridge but today's news is positive."
Foster also responded to Rennie's threat to skip the first test altogether, saying there's now no excuses for Australia not to play both tests.
"He's come out strong based on information about the quarantine. We listened to that. But at the same time we were busy talking hard with the Government about relaxing it.
"Now I think we've achieved that, when you look an equitable situation which is about giving two teams a fair chance at preparing. To basically have 13 free days to prepare for a test match, in an international sports world is huge.
"Really, I don't think there's any excuse for them not to come over and play in that weekend."
New Zealand was confirmed to host two Bledisloe Cup tests after missing out on the hosting rights for the Rugby Championship due to restrictive quarantine protocols.
Foster admitted losing the Rugby Championship hosting rights to Australia was disappointing but refused to say New Zealand "blew it".
"I wouldn't say we blew it. It looks like it because we had it and then it went. While it's frustrating, we're fully aware there are a whole lot of variables in play and quite frankly life hasn't been fair this year for too many people and we've just got to take that one on the chin.
"What we do need to do is make sure we do these two Bledisloes and in a fair way. I think what's happened now with the quarantine has basically put us in a situation where we can offer the Aussies nine full days in quarantine preparing, four full days once they are out, which is about 13 days by my maths.
"When you look at our preparation for the first Bled, we've got a three-day camp, a four-day camp and a six-day lead-in – so we've got 13 days. Really can't see any reason we can't be playing on that weekend."
No official dates have been confirmed for the Bledisloe Cup tests yet.