England and Scotland meet in the Euros at Wembley on Friday.
"Billy's a brilliant player," said Mount. "I was always interested in his development and obviously seeing him make the step up and playing in the first team and seeing how well he's been doing, it's been brilliant to see.
"I've known him for a long time, since he first joined Chelsea. I met him when he first joined, and then I went out on loan and he was playing in the youth teams, but I always tried to keep an eye on him and what he was doing because he was a young midfielder coming through that I'd heard a lot about."
Gilmour has become a popular member of the first-team squad at Cobham, with Mount confirming the friendship among the Chelsea players will cross the divide of that international rivalry.
"I'm very close with him, along with Chilly as well,' he added. 'We're close and he's doing really well and I'm obviously looking forward to playing against him."