Passan added that the contract includes a fourth-year option. The deal will see Kimbrel paid $10 million this season, and $16 million in 2020 and 2021. His physical is slated for Thursday.
Kimbrel, who turned down a $17.9-million qualifying offer in November from the World Series champion Boston Red Sox, was one of the last top free agents available.
The seven-time all-star topped the National League in saves in both 2013 and 2014. He tied for the lead in 2011 and 2012.
The 31-year-old has collected 333 saves across nine seasons. He spent the last three campaigns with the Red Sox after one year in San Diego and five in Atlanta.