The second-row, who can also play in the back-row, has agreed a two-year deal subject to a medical.
The 25-year-old joins from the Fiji Barbarians rugby club and has 19 caps for his country.
Nakarawa becomes the third Fijian at the Warriors, joining compatriots Jerry Yanuyanutawa and last year's Player of the Season, Niko Matawalu.
He arrived in Glasgow on Tuesday and is expected to join the club's pre-season training on Wednesday.
Nakarawa said: "I am very excited to have signed my first professional contract with Glasgow Warriors.
"I'm happy to be here and I'm looking forward to joining the squad for pre-season training.
"I've always wanted to play rugby outside of Fiji and now I have that opportunity.
"I could have moved to England or France last year but stayed in Fiji. I am determined to make the most of this move.
"I've heard a lot about Glasgow from Niko and he talks very highly of the club. This will be a new experience for me and I can't wait to get started."
Glasgow head coach Gregor Townsend said: "This is excellent news for the club.
"We've been tracking Leone for a long time and we're delighted that he's agreed to join us.
"He has played both fifteens and sevens for Fiji and is a dynamic and skilful player. His ball carrying ability and versatility will really strengthen our squad.
"Leone played for Fiji at the Rugby World Cup Sevens at the end of June, so we expect his fitness to be at a high level.
"We now have great competition for places in our forward pack, with a mix of different qualities, and this should stand us in good stead next season.
"We're looking forward to Leone starting pre-season training with the rest of the squad."