United are already 12 points off the pace at the top of the table and the club have announced their intention to appoint an interim manager until the end of the season before making a permanent appointment.
Here, AFP Sport looks at five of the leading candidates to be Solskjaer's successor:
Expectations were high for United to challenge for major trophies this season after the signings of serial winners Cristiano Ronaldo and Raphael Varane, along with £73 million ($100 million) winger Jadon Sancho.
Three of Ronaldo and Varane's Champions League titles came under Zidane at Real Madrid and the Frenchman is also a free agent having ended his second spell with the Spanish giants in May.
Zidane proved himself a master of moulding a star-studded squad into a team at Madrid, where he also won two La Liga titles.
That is exactly what United need and his global profile fits the brand-building exercise the English giants have gone down in recent years, at times to the detriment of a well-functioning team on the field.
According to reports, he is the favoured candidate of the club's American owners, the Glazer family.
The outstanding option on the domestic market for United is Leicester boss Brendan Rodgers.
The Northern Irishman's past as a Liverpool manager may rule him out for many United fans, but he has rebuilt his reputation since a rollercoaster three years at Anfield between 2012 and 2015.
Rodgers proved himself capable of handling the demands of winning every week in three years at Celtic, where he won a clean sweep of seven domestic trophies before returning to the Premier League with Leicester.
The Foxes have narrowly missed out on the Champions League for the past two seasons to Chelsea despite far inferior resources and beat the European champions to win the FA Cup for the first time in the club's history last season.
However, Leicester have also struggled of late with just two wins in their last nine league games.
Erik ten Hag
The Dutchman is responsible for one of the form teams in Europe right now with Ajax already into the Champions League last 16.
Ten Hag was linked with a number of top jobs after leading the four-time European champions to the semi-finals of the Champions League for the first time since 1997 two years ago.
But unlike many of his players from that run, he is yet to be poached by one of Europe's top clubs.
The former Bayern Munich youth team coach would fit the attacking style demanded of a United manager.
The Spain coach is another with plenty of experience in getting the best out of a collection of talented individuals.
He won the treble in his first season at Barcelona and added another La Liga title with a front three of Lionel Messi, Luis Suarez and Neymar.
Fluent in English, Luis Enrique has never hidden his desire to one day coach in the Premier League.
His success with Spain this year at Euro 2020 and the Nations League finals shows he is also capable of producing an exciting side without the same quality of stars he had on offer at the Camp Nou.
The Argentine is the one that got away for many United fans as the Red Devils kept faith with Solskjaer while Pochettino was a free agent for over a year following his sacking by Tottenham in 2019.
Now in charge of Paris Saint-Germain, Pochettino is unlikely to jump ship mid-season with a 12-point lead at the top of Ligue 1 and another chance to win the club's first ever Champions League.
However, Pochettino's time in the French capital has not been plain sailing, particularly in his dealings with PSG's sporting director Leonardo.
By waiting until the end of the season to make a permanent appointment, United are opening the door to another shot at luring Pochettino to Old Trafford.