Kagawa completed a €16.4 million move from Borussia Dortmund last week and said of the shirt number vacated by Michael Owen: "Since I am very new to the club, I have no intention to get a 'big' number."
The 23-year-old instead allowed winger Antonio Valencia to take the famous shirt and inherited the No.26 most recently worn by Gabriel Obertan ahead of his first season.
And Robson, who wore Valencia's new number during a distinguished spell at Old Trafford, believes the move says a lot about Kagawa's personality.
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"One thing you get with Japanese boys is they’re very hard working and conscientious about the job."
Kagawa was part of a Borussia Dortmund side that sealed a League and Cup double last season playing as a central attacking midfielder in a 4-2-3-1 system.
And, though there is much debate over where Kagawa will feature for the Red Devils, Robson feels the 23-year-old has the versatility to slot in anywhere.
He added: "I think the boss will be looking at where he does play him out there on tour. He’s got a good option now and can play Javier Hernandez and Danny Welbeck up top but can also now play Wayne [Rooney] up there if he really wants to and Shinji can play in behind him.
"From what I’ve seen and been told about him, he can play in numerous positions and won’t let you down. He can do a good, hard, workmanlike job for you.
"Versatility has always been important for the boss. He has a few players who can play in different positions.
"Look at his last few signings – even at the back, Phil Jones and Chris Smalling can play centre-half or right-back. Phil can also fill in to midfield as well so the boss likes players who have got that flexibility where they can balance a team."