The deal is pending a physical.
Toronto acquired the 27-year-old right-hander at the trade deadline from the Minnesota Twins, and he went 5-4 with a 3.58 ERA in 12 starts for the Blue Jays down the stretch.
Berrios went 12-9 overall in 2021 as he set career bests with a 3.52 ERA and 204 strikeouts. He is 60-47 with a 4.04 ERA in his six-year career.
Berrios, a two-time All-Star, was scheduled to become a free agent after the 2022 season.
Meanwhile, the Jays' staff ace, Robbie Ray, is currently a free agent. The American League Cy Young finalist went 13-7 with a 2.84 ERA and led the major leagues with 248 strikeouts.
Left-hander Steven Matz, who went 14-7 with a 3.82 ERA last season, also is a free agent.
ESPN's Jeff Passan contributed to this report.