There is again speculation surrounding Ten Hag's future due to the team's issues this season, with the Dutchman in a difficult position following Sunday's 3-0 loss to Tottenham Hotspur in the Premier League.
Ten Hag will allegedly be given the next two games against Porto and Aston Villa, but defeats in both of those fixtures could see him removed from his position ahead of the October international break.
If that happens, Ruud van Nistelrooy would be the favourite to take the job on an interim basis, with the former Red Devils striker returning to Old Trafford over the summer as an assistant manager.
However, according to inews, Brailsford, who is now a senior figure at Man United as part of his role as director of sport at INEOS, would push for Southgate to be appointed as the club's new permanent head coach.
Brailsford 'would push for Southgate arrival' at Man United
The report claims that Brailsford's relationship with Southgate has strengthened since the latter left his role with England following the completion of Euro 2024.
Southgate led the Three Lions to the final of the European Championship, and the Football Association had been keen for him to remain in charge, but he decided to stand down from his role.
Not all of Man United's leadership team are said to be convinced by Southgate, but Brailsford would allegedly fight for the 54-year-old to be appointed as the club's new manager if a change is made.
Southgate was in charge of England between 2016 and 2014, and his only experience of club management came between 2006 and 2009 at Middlesbrough, where he boasted a record of 45 wins, 43 draws and 63 defeats from 151 games.
Who else would be in the frame to replace Ten Hag?
Van Nistelrooy is the obvious choice, with the Dutchman a ready-made replacement considering that he is already at the club and knows Man United well following his time as a player at Old Trafford.
Thomas Tuchel is also still in the picture, with the German leaving Bayern Munich over the summer, while Graham Potter also cannot be completely ruled out at this stage of proceedings.
Ole Gunnar Solskjaer has opened the door to a spectacular return, but it is highly unlikely that the Norwegian would be considered for the role.
Aside from Van Nistelrooy and Southgate, Tuchel is perhaps the only realistic target, although Ipswich Town's Kieran McKenna has admirers at the club and could emerge as a candidate if there is a change in the near future.