Suzuki, 35, would get $4 million next year and $6 million in 2020, up from $3.5 million last season, his second with the Atlanta Braves.
The deal was first reported by MLB.com.
Suzuki played for the Nationals for parts of the 2012 and '13 seasons, hitting .239 with 8 home runs and 50 RBIs in 122 games. He was acquired by the Nationals from the Oakland Athletics in 2012 and then traded back to the A's in 2013.
Suzuki is coming off the best two offensive seasons of his career and is strong defensively. He hit .271 in 2018, and he had 31 home runs with 100 RBIs in 186 games over his two seasons with Atlanta.
Suzuki, who has a .258 career batting average, has played with Minnesota, Washington, Oakland and Atlanta over his 12 seasons. He was named to the 2014 American League All-Star team while with the Twins.
He has thrown out 219 baserunners, which ranks fourth among active catchers. He also ranks fourth in games caught (1,229) and putouts as a catcher (8,250).
Veteran catcher Matt Wieters, who appeared in 76 games for the Nationals during an injury-marred 2018 season, is a free agent.
The Associated Press contributed to this report.