Round believes Giggs should be handed the role.
When asked if Giggs would be the correct choice, Round said: “I would hope so, I think that is the right and proper succession plan for Manchester United.
“With us he was part-coach, part-player and now he has stepped into being an assistant manager – the natural progression for him is to go on to be a manager.
“You would assume that would be at Manchester United; that is what they are grooming him for and getting him ready to take over eventually when that time comes.
“I think that is only right and proper – he needs and deserves to have that opportunity.
“Manchester United, to move forward, need to give him a chance to either succeed and thrive or fail and move away.”