Leiua has played 30 games for Wasps since joining them from Super Rugby side Hurricanes in April 2014.
The 28-year-old, who can also play on the wing, has scored two tries in 11 Test appearances for Samoa.
"He's a robust midfielder who can play in a number of positions in the backline," Bristol head coach Mark Tainton told the club website.
"Alapati was a standout performer for the Hurricanes in Super Rugby and made a similar impact in his first season with Wasps - he will be a fantastic addition to the Bristol Rugby squad."
Bristol are currently bottom of the Premiership table, one point behind 11th-placed Worcester with nine games left to play this season.
Bristol Rugby have signed Wasps' Samoa international centre Alapati Leiua in a deal to start from next season.