The deal is reportedly a four-year, £16m contract that will keep Landry with the Colts until he is 32.
"We're ecstatic about this signing," Colts general manager Ryan Grigson said in a statement. "We feel LaRon is an absolute game changer and a true impact player.
"We're talking about a 220-pound safety that runs 4.3 and plays to that speed. His approach to the game and his style of play are lights out and embody the culture we're building on the defence and this team in general."
Landry made 100 combined tackles last season for the Jets.