An announcement from the club is expected later today, it is understood.
United will also try to move quickly to install a permanent replacement for Moyes. It is understood that Holland boss Louis van Gaal is the favourite to take over, but another potential candidate in Borussia Dortmund manager Jurgen Klopp looks to be out of the picture.
United are determined not to make the same mistake as last summer when Moyes did not take over until the start of July, meaning a big chunk of the transfer window had already passed.
Van Gaal's credentials make it clear why he is the clear front-runner.
He has won Spanish league titles with Barcelona, and with Bayern Munich he reached the 2010 Champions League final as well as winning the Bundesliga. With Ajax, he won the Champions League in 1995.
He is also used to working with big-name players and attracting big signings - neither of which were part of Moyes' track record.
Van Gaal has also made it clear that he will be looking to move on from being Holland head coach after the World Cup.
Klopp has been touted as a candidate for the top jobs in European football, most recently as the next Barcelona head coach, but he has made it clear he is staying at Dortmund.
Bookmakers have slashed the odds on Van Gaal from 8-1 to clear 11-4 favourite.
Ladbrokes odds: 11-4 Louis Van Gaal, 4-1 Jurgen Klopp, 6-1 Ryan Giggs, 12-1 Pep Guardiola, 14-1 Diego Simeone, 16-1 Jose Mourinho, 16-1 Roberto Martinez, 16-1 Sir Alex Ferguson, 16-1 Frank De Boer, 16-1 Carlos Queiroz.
Klopp wasted little time in ruling himself out of the Manchester United job.
The 46-year-old said: "Man United is a great club and I feel very familiar with their wonderful fans. But my commitment to Borussia Dortmund and the people is not breakable."
Klopp agreed a new contract at the Bundesliga club until 2018 back in October and said last week that he was "still in love" with Dortmund.
Ryan Giggs timeline
1973: Born Ryan Joseph Wilson on November 29 in Cardiff.
1990: Signs as a trainee with Manchester United in July and as a professional in December.
1991: Adopts mother's maiden name and becomes Ryan Giggs.
March - Makes United debut aged 17 as a substitute against Everton. Two months later he scores the winner against Manchester City at Old Trafford.
October - Becomes the youngest footballer to represent Wales when he plays against Germany in Nuremberg aged 17 years and 322 days.
November - Wins his first piece of silverware with United as the Reds beat Red Star Belgrade to lift the UEFA Super Cup.
1992: Helps United lift League Cup and wins PFA Young Player of the Year award.
1993: Becomes first player to win PFA Young Player of the Year award twice.
Member of United side which ends the club's 26-year wait to win the league title.
1994: Helps United win the Premier League and FA Cup double.
1996: Helps the Red Devils win the double for a second time.
1997: Wins a third league title in four years and in October scores a memorable goal in 3-2 Champions League victory over Juventus at Old Trafford.
1999: Scores stunning winner in United's epic FA Cup semi-final replay victory over Arsenal at Villa Park in April. Helps the Reds complete an historic treble of Champions League, Premier League and FA Cup.
2000: Wins his sixth Premier League winners' medal in eight years.
2001: Collects his seventh league championship and 12th major honour in total.
July - Signs a new five-year contact. Sent off for the first time in his career as Wales crash 3-2 in Norway.
2002: August 23: Nets his 100th career goal for United with the team's second goal in league game at Chelsea.
2003: May - United are crowned league champions again.
2004: May - United finish third in the table but Giggs lines up in the 3-0 FA Cup final victory over Millwall.
2007: May 6 - Wins ninth Premier League title with Manchester United.
May 30 - Announces retirement from international football.
December 11 - Receives an OBE from the Queen for services to football.
2008: May 11 - Equals Sir Bobby Charlton's appearance record for the club when he plays his 758th game, against Wigan.
May 21 - Breaks Charlton's record after coming on as a substitute - and scoring a penalty in the dramatic shoot-out - as Manchester United beat Chelsea in the Champions League final.
2009: April 26 - Voted Professional Footballers' Association player of the year.
May 16 - Helps United win the Premier League and becomes the first footballer to collect 11 top division English league title medals.
December 13 - Named BBC Sports Personality of the Year.
December 31 - Named the Manchester United Player of the Decade.
2010: February - Wins the League Cup as United beat Aston Villa 2-1 in the final.
2011: January 17 - Makes his 600th league appearance for United, against Tottenham at White Hart Lane.
March - Surpasses Charlton's Manchester United league appearance record by playing his 607th game against Liverpool.
May - Wins the Premier League title for a 12th time.
2012: February - Makes his 900th appearance for United, scoring a last-minute winner in a 2-1 away win against Norwich.
June - Named as one of three over-age players in the Great Britain squad for the Olympic Games.
August - Team GB knocked out of Olympics in quarter-final by South Korea on penalties.
2013: February 10 - Scores opening goal in United's 2-0 win over Everton, continuing his record of scoring in every season of his professional career.
(At April 22, 2014, he has yet to score in the 2013/14 season.)
March 5 - Makes 1,000th senior competitive appearance, turning out for United as they controversially crash out of Champions League against Real Madrid at last-16 stage.
May - Lifts Premier League trophy for a 13th time.
July 4 - United confirm Giggs will take up a player-coaching role from 2013/14 season, under new manager David Moyes.
2014: April 22 - Following the sacking of Moyes, Giggs is poised to be announced as interim manager.