The 28-year-old will sign a one-year contract subject to a medical and a visa.
Yanuyanutawa, who played 21 times for London Irish last season and has 10 international caps, follows South African flanker Tyrone Holmes in agreeing a move to the Warriors this week.
The former Brumbies prop will play alongside Fiji team-mate Niko Matawalu.
"I've spoken to Niko about the move and he says it's a good club, with a great bunch of boys and a good coaching system," he said.
"He's really enjoying it so I'm sure I will too.
"It was good to win the IRB Pacific Nations Cup for the first time with Fiji last month and signing for Glasgow has capped off a good year for me.
"I've always wanted to play in the Heineken Cup and I didn't get that opportunity last season with London Irish.
"It's the pinnacle of European rugby, like being in Super Rugby in the southern hemisphere. To get the opportunity to play against Toulon in the Heineken Cup next season would be unreal."
Glasgow head coach Gregor Townsend added: "He is an extremely powerful and mobile prop, who will give us another option in the front row."