But chairman Bill Kenwright says he will not make a snap decision.
"Eleven years ago I made a decision [to appoint Moyes] and it was an instant decision. I don't think that can happen this time," he said.
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"We have to see what candidates are out there to take the club forward."
Laudrup, 48, led Swansea to the League Cup trophy this season - the club's first major silverware - while Martinez, 39, has been at Wigan since 2009.
Departing captain Phil Neville, 36, may also figure in Everton's plans.
Kenwright will take on board the views of the fans before making an appointment and said: "I will be looking to the fans to get that guidance.
"I can't individually poll each one of them but it is important that they get the right manager.
"The fans know the adventure they have had for 11 years. A very important part of my life is to see they are not let down because I don't want Evertonians let down."
Kenwright praised Moyes for his contribution at Goodison Park over the last 11 seasons.
"Manchester United are very lucky," he said. "It will be tough for all Evertonians to say goodbye to him, a great manager.
Everton were shocked by the speed of events that have taken David Moyes to Old Trafford. The manager was planning for next season and even held meetings about transfer targets earlier this week before Sir Alex Ferguson's retirement. There was a quiet confidence behind the scenes that, in the apparent absence of attractive offers, Moyes would stay at Goodison Park after his contract expires this summer.
Now owner Bill Kenwright must find a successor to the man who has led Everton for 11 years.  Swansea's Michael Laudrup and Wigan Athletic manager Roberto Martinez head the list but other candidates will merit discussion, such as departing Everton captain Phil Neville. If the decision on who succeeded Ferguson was crucial to Manchester United, the same can be said for Everton as they seek to replace the manager who has been central to the workings of the club since 2002.
"We could not stand in his way because he was out of contract. It was his decision; he has made it."
Neville, who may yet emerge as a candidate to join the new managerial team at Old Trafford, announced last month that he would leave Goodison at the end of the season.
He is highly respected at the club and will gain coaching experience with England at the European Under-21s Championship in Israel this summer.
Former Barcelona and Real Madrid midfielder Laudrup started his managerial career at Danish club Brondby, guiding them to the title and Danish Cup twice during four years in charge.
He then had spells at Spain's Getafe and Spartak Moscow in Russia, before being appointed Real Mallorca manager in La Liga in July 2010.
He joined Swansea City last summer, replacing Liverpool-bound Brendan Rodgers, and has enjoyed a successful first season in England, guiding the Welsh side to their first ever major trophy and ninth place in the Premier League.
Laudrup still has 14 months left on his contract but, with the Dane also linked to Real Madrid, Swansea chairman Huw Jenkins has admitted he has a plan in place should he leave.
Former Swans manager Martinez was a target for Aston Villa in June 2011 but opted to stay as the Latics manager, and was also strongly linked with the Liverpool job last summer.
In his first managerial job, the ex-Real Zaragoza and Swansea midfielder guided the Swans into the Championship in 2008.
He was appointed Wigan boss in 2009 and has since helped keep the north west club in the Premier League, while managing on a low budget compared with his rivals.
But Martinez, whose side face Manchester City in the FA Cup final at Wembley on Saturday and are in a battle to avoid Premier League relegation, said: "It would be a waste of time for anyone [to talk about it] at the moment.
"The most important thing for me is to be ready for Saturday. This is the peak of our season and we are not going to lose any focus or concentration."
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Former Manchester City, Blackburn Rovers, Fulham and QPR boss Mark Hughes, 49, who played for Everton between 2000 and 2002, refused to rule himself out of the running to replace Moyes.
"It's not happened yet but it's obvious if one manager leaves there is an opportunity for other managers who are currently out of work, which includes myself," he told Sky Sports News.
Meanwhile Moyes has paid tribute to the Everton board, players and fans.
He said: "I have had a terrific job at Everton, with a tremendous chairman and board of directors and a great set of players.
"Between now and the end of the season, I will do everything in my power to make sure we finish as high as possible in the table.
"Everton's fantastic fans have played a big part in making my years at Goodison so enjoyable and I thank them wholeheartedly for the support they have given me and the players.
"Everton will be close to me for the rest of my life."
Swansea's Michael Laudrup and Wigan's Roberto Martinez are the frontrunners to replace David Moyes as Everton boss when he takes over from Sir Alex Ferguson at Manchester United in July.