The Red Devils will be under the tutelage of Ralf Rangnick for the remainder of the 2021-22 season before the German helps pick out a permanent manager from his consultancy role.
"It is a shame for Manchester United that Brendan Rodgers once managed Liverpool. Otherwise I'd see him as the ideal fit to mastermind the long-term project at Old Trafford," Murphy wrote in the Daily Mail.
"Past associations don't have to be a deal-breaker but they don't help, as Rafa Benitez is finding out at Everton. So Mauricio Pochettino looks the favourite to be in charge next season unless Ralf Rangnick does so well that he has to stay.
"Rodgers ticks so many boxes: he has extensive experience but is still young enough to have the hunger and drive to take United forward. Rodgers knows what it's like to handle the spotlight after managing Liverpool and Celtic.
"It's exciting to think what he could do with the squad at Old Trafford because his track record at improving players is brilliant."