Ben Volin of the Boston Globe reports Brissett's deal is worth $5 million guaranteed with the potential to make $7.5 million with incentives.
Per NFL Network's Tom Pelissero, Brissett will serve as a backup and mentor to quarterback Tua Tagovailoa.
The move comes after veteran quarterback Ryan Fitzpatrick agreed to a one-year deal with the Washington Football Team.
Brissett is coming off a season serving as Philip Rivers's backup but did not log much playing time. The signal-caller came up clutch for Indy during the 2017 and 2019 seasons while Andrew Luck was sidelined before choosing to retire.
He ended the 2017 season with 3,098 passing yards, 13 passing touchdowns, seven interceptions, 260 rushing yards and four rushing touchdowns. In 2019, Brissett totaled 2,942 passing yards, 18 passing touchdowns and six interceptions.
In his five-year NFL career with stints in New England and Indianapolis, Brissett threw for 6,459 yards, 31 touchdowns and 13 interceptions in 49 games.