Moyes has emerged as the leading candidate to replace Sir Alex Ferguson at Old Trafford following his fellow Scot's announcement that he will retire at the end of the season.
Ferguson will take up a role as a director and ambassador, and reports have suggested he has played a key role in identifying his managerial successor.
Sky Bet suspended betting on Moyes heading to United and he held talks with Everton chairman Kenwright on Wednesday evening in London.
Following the discussions on Wednesday evening, Kenwright confirmed that no agreement has been finalised for Moyes to succeed Ferguson. Moyes also refused to speak to waiting reporters after the meeting with his chairman.
Kenwright also confirmed to Sky Sports News that Moyes will be in charge for Everton's next Premier League game against West Ham United on Sunday.
"I don't know about a deal being done," Kenwright told Sky Sports News. "David had a meeting scheduled with me and we've just had a good meeting.
"I think, at the end of the day, David's contract is up in six weeks' time and he has a right to make his own decisions.
"He's served this club wonderfully well."
The 50-year-old former Preston North End boss has spent 11 years in charge at Everton, but is out of contract at the end of the season.
United chief executive David Gill has outlined the qualities needed to succeed Ferguson as manager amid expectation Moyes will be the new man in charge.
"The qualities are the ones that have been inherent at Manchester United for many years," Gill told MUTV.
"Our two most successful eras were with managers who got involved with all aspects of the club, from the youth team to the first team, to get that degree of loyalty and understanding of the football club.
"Clearly he has to have the requisite football experience, both in terms of domestic and European experience. It is a small pool."
Former Manchester United striker Michael Owen said he would choose the Everton boss.
Owen, the former Liverpool striker who played under Ferguson for three seasons, said on Twitter: "What an act to follow. I guess only Mourinho would have the confidence to strut through the door. If it were me, I'd go for Moyes."
Former United keeper Peter Schmeichel also believes Moyes would be a good fit as a successor as he has proved he can stay at one club for a long time, spending 10 years at Everton.
"What we are looking for is, not someone to come in 10 months or three years, we want someone to come stay there and give stability," he added.
"When we talk about Moyes, he has been a decade at Everton and done a fantastic job on limited funds.
"Is he cut from the same cloth? I don't know. Don't forget there will be major changes with David Gill stepping down - the club is really changing in the summer, the more stability they can add the better.
"David Moyes, he sounds like a good solution."
Sky Sports pundit, and former United captain, Gary Neville believes that whoever does succeed Ferguson has a perfect opportunity to build on the Scot's legacy and will be guaranteed the support of the fans.
"The club will always move on - Manchester United is a massive football club," said Neville. "It moved on from the Munich air disaster, it moved on from Sir Matt Busby, it will move on from Sir Alex Ferguson.
"People will say 'how will the new manager be able to replace him?' and there will be comparisons. It's quite simple - play attractive football, bring young players through and win a championship.
"The fans will support them, the club will support them. This is not a stupid football club, this is not a crazy football club that changes the manager every 10 minutes.
"The new manager will need time and the club will need time to move on but they've got sensible people there.
"Somebody will be extremely fortunate to follow him. They are going into a football club where so much is perfect. The staff have been there for 15-20 years - they are grounded in winning, they have a mentality for winning and how to behave. It really is fantastic."