Scottish teenager Gilmour was thrust into action in an FA Cup victory over Liverpool in March and followed up a Man-of-the-Match display in that game with a fine showing in a 4-0 hammering of Everton.
The suspension of the Premier League due to the coronavirus pandemic stemmed the 18-year-old's promising momentum but his performance against Liverpool looked "world class" according to Keane, a man notoriously difficult to impress in his role as a pundit.
Blues boss Lampard was surprised to hear such effusive praise from Keane but believes Gilmour was deserving of such comments.
"I had to put my cup of tea down to hear Roy say something nice!" Lampard said.
"I'm only joking – Roy was a fantastic midfield player and he probably won't give credit unless it's due. It certainly was due.
"I heard the statements he made about Billy and he was spot on, and Billy should be happy with that. If he goes and follows the ideas of midfield play that someone like Roy Keane says, he won't go far wrong."
Keane was part of a legendary midfield at the Red Devils alongside Scholes. Lampard was keen to temper expectations but did concede Gilmour possesses similarites with the latter.
"There aren't many days when you don't see him show up and try to do the right things," Lampard added.
"Paul Scholes was obviously one of the greatest midfield players I've ever played against in the Premier League.
"It was interesting to play against Scholesy: in the early parts of the career when he was a goal-getter, he'd play it around the corner and then would arrive in the box and have the ability and the feet and the nous to score the goals or make assists regularly, and then to see him change his career [was impressive].
"The cut-throat part of the pitch is probably at the top end, where it becomes much tighter and quicker.
"In midfield, it can be easier because you have more time in those deeper areas. But if you're a player that's not top quality like Scholesy, more time can sometimes be a problem for you.
"He would always pick the right pass, he had incredible range, and if you did try to jump from midfield to get close to him, he'd just play it round the corner because he'd know you were coming.
"Billy has shown early signs of that sort of ability. I'm not comparing him to Scholesy – no way – but in terms of those basics, Billy certainly has them and that's a great platform to build hopefully a fantastic career."