Brighton midfielder Gilmour, 22, has impressed for the Seagulls this season and helped his country to qualify for Euro 2024.
Former Aston Villa and Tottenham defender Hutton, 38, spoke highly of Gilmour during an interview with sports betting site OLBG.
Hutton said: "It’s taken a little bit of time to kind of show his qualities. He’s not just tidy on the ball or someone who just keeps the ball ticking over, which at times I thought he was. I think he's now added to his game going forward and he's tenacious.
"I didn't realise he was like that. He wants to win the ball back. He can be aggressive and he very rarely gives the ball away. I think those are top qualities to have as a midfielder. He stands out for Scotland."
Hutton, who won the Scottish Premier League with Rangers in 2005, said Gilmour is playing ‘at a very high level’ for Roberto De Zerbi’s Brighton.
"He's still very young and he's learning from a top-quality manager,” he said.
"So I think he's only going to get better and his confidence will grow. In a Scotland shirt, he’s a key part of what we’re doing at this moment in time. I think he really does stand out for Scotland with the qualities he has."