But it is Moyes' appointment that looks the most demanding, as he steps into the shadow cast by Sir Alex Ferguson.
Graham told the Mirror: "It is the natural progression in his career. He has served his time as an apprentice at Preston and then at Everton - where he did a brilliant job. I believe he is the perfect choice to take over.
"United could have turned to a foreign manager, but I am glad they stayed British.
"There are a lot of similarities between Moyes and Sir Alex.
"Moyes has the same hunger and ambition - and he has that drive that is so common from people from the west of Scotland.
"He has drive to be successful. He is also a realist and knows there will be inevitable comparisons between him and Sir Alex, especially if United lose a few games.
"But he will ready for that. He will be prepared - and it will not concern him about handling big names.
"There were star players at Everton when he first arrived. It won't be a problem for him. He will relish the challenge and it would be no surprise to me if he won the title next season."