Fresh from leading Ajax to his third Eredivisie title as manager, Ten Hag was appointed by the Red Devils in April and has the task of turning things around at Old Trafford.
Things look pretty bleak at Man United at the moment with the club ending the season with the club's worst ever Premier League points tally.
Their underperforming squad is also in need 'open heart surgery', according to outgoing interim boss Ralf Rangnick.
Ten Hag comes with a big reputation, having worked under Pep Guardiola at Bayern Munich, but has not managed a senior side outside the Netherlands.
Southgate has won plenty of plaudits for the work he has done with England, leading them to a World Cup semi-final and then the final of Euro 2020.
The former Middlesbrough boss still has his detractors, but ex-Liverpool and Three Lions midfielder Murphy has been impressed with how Southgate has developed.
When asked who he would pick for the Manchester United job, Southgate and assistant Steve Holland, or Ten Hag, Murphy said: "Gareth and Steve."
When pressed on why, he said: "I know them. I think they've come a long way, the pair of them, and have a great partnership.
"What they've achieved with England, managing big players. Gareth has evolved into a very intelligent football manager.
"With Steve by his side, they would do a great job at a big club.
"What United need now isn't all about on the pitch. They will be judged on that, but initially what do United need?
"They need someone to bring together this squad and try and create a unity, atmosphere and energy that England have. I think that would be fair.
"What Ten Hag has done, of course it is credible winning the league three times with Ajax and he did alright with Utrecht.
"He's not managed big players at a big club, so we don't know how he's going to deal with that pressure.
"Every part of his life will be magnified, whether that will be the wrong shop at the wrong time, or drives the wrong car. Everything he does, is now going to be talked about."