Royle was recruited as football consultant by the relegated Canaries on Monday, with Gary Holt and Mark Robson named first-team coaches in Neil Adams' backroom team.
Norwich are planning for life in the Sky Bet Championship after slipping out of the Premier League following a three-year spell in the top flight.
And Royle admits Snodgrass, who has been targeted by the likes of Celtic and Hull City and only has a year left on his current contract, is the kind of player Norwich need to keep at Carrow Road.
The Scotland midfielder scored six goals in 29 league appearances last season and Royle said: "We are all hoping he will be here next year. He’s got a great attitude and we think very highly of him.
"I'm told that right to the last whistle of the last game he was doing his best to win a game for Norwich. He’s the type you want around."
"It would be nice if we could keep the majority of players that Neil knows but there will inevitably be one or two departures and there will be two or three incoming players," Royle added on talkSPORT.
Russell Martin, meanwhile, admits he would be surprised if no-one comes in for Snodgrass in this summer's window.
He said: "I'm sure there will be Premier League clubs who want some of our players. Some of them have shown they can do a job at that level for the past two or three years.
"If that happens, which it will and it will also happen at the other relegated clubs, then it is up to the board to do what is best.
"In terms of Snoddy, I would be surprised if no-one came in for him."