Chiefs head coach Rob Baxter has confirmed that Tui will join current French second division leaders Lyon on a two-year deal.
Tui, 29, has made more than 100 appearances for Exeter since joining them from Italian outfit Calvisano in 2010.
"I have no problem with a player moving on for positive reasons," Baxter said.
"Hoani is moving on - not because we don't want him to stay - but more that a very good French club is prepared to offer him a very good contract.
"He has been a great player for us, he's a good guy around the place and he's continued to perform very well for us even since we knew he was going to leave."