United vice-chairman Ed Woodward is leading the search but is drawing on the experience of Sir Alex Ferguson and former chief executive David Gill, who are both board members.
Holland coach Louis van Gaal is the front-runner, but sources close to United say he is only one on a list of candidates that has been drawn up.
The club are looking for a coach with a significant level of experience, which means temporary manager Ryan Giggs will not be among those considered.
Van Gaal is currently on holiday in Portugal, according to reports in Holland, and the 62-year-old will be a free agent after the World Cup.
The Dutchman is one of those on the shortlist, along with Real Madrid coach Carlo Ancelotti.
However, former Real Madrid president Ramon Calderon is confident Ancelotti will stay in Spain.
He said: "I believe Carlo Ancelotti will be with Real Madrid next season. He's got a contract. We don't know what's happening with Manchester United.
"It's always a pity when you have to sack a coach. I had that problem with Fabio Capello. Sometimes you have to take those decisions, though. Manchester United's results have not been normal so it's a decision they had to take."
As well as a significant level of experience, the criteria for the new manager is also thought to include having a good grasp of English, which would count against Atletico Madrid's much-admired coach Diego Simeone.
United are trying to move on from the controversy that surrounded the sacking of Moyes on Tuesday, and Giggs will hold his first news conference as player-manager on Friday morning.
The 'Chosen One' banner that has been displayed on the Stretford End since Moyes took over has been taken down, it has been confirmed.
It is also understood that club chiefs will continue to work hard to line up transfer targets so that the deals can be done as soon as the transfer window opens on June 1. That will also give time for the new manager to indicate his views on the targets as well as put forward other transfer wishes.
United have refused to comment on reports that the owners, the Glazers, are due to fly into the UK for talks on the situation.
Meanwhile, West Ham manager Sam Allardyce claimed the process of Moyes' departure could have been handled better.
He said: "For Manchester United as an institution and a football club as large as they are, the process looked like it was flawed, which was a great shame for Manchester United as a brand and, of course, David Moyes knowing his fate before he was even told.
"That is how it looks from the outside. If that is the case, it is a great shame for David and a great shame that Manchester United chose to use that process."
United have insisted no decision had been taken at the time reports first started emerging that Moyes faced the axe.
Roberto Martinez is confident that Moyes, his predecessor at Everton, will have no problems in getting a new job.
He said: "It is not a nice situation when a manager loses his job, especially someone who has such strong links with our club.
"It has been a difficult experience but any experience is a good experience in this game unfortunately.
"We know David Moyes will look forward to the next footballing chapter.
"I am sure he will be ready to get into another job as quick as he can and I am sure he won't have a lack of offers or opportunities when you have someone who is so hard-working, focused and dedicated in his job."