The financial details were not disclosed by either side, but British media reported that the buy option is worth around 40 million euros ($43.66 million).
Todibo, who has been capped twice for France, started his career at Toulouse before moving to LaLiga side Barcelona in 2019. He made 136 appearances for Nice since his 2021 switch.
"This is an amazing opportunity for me, at a club with huge ambition, who want to really make progress under the new head coach," Todibo, who was earlier linked with a move to Italian giants Juventus, said in a statement.
"It feels like the right time to be coming to West Ham, in the heart of London, with an amazing, passionate fan base. I'm so excited to pull on the West Ham shirt, and play at the London Stadium in the Premier League."
West Ham, who finished ninth last season, begin their Premier League campaign at home against Aston Villa under new manager Julen Lopetegui on August 17th.