The deal includes options for 2025 and 2026 and could be worth $29.25 million over four seasons.
Green is likely to miss at least the early portion of next season while recovering from Tommy John surgery. He injured his pitching elbow last May with the New York Yankees and was sidelined the rest of the season.
The right-hander got off to a solid start in 2022 with a 3.00 ERA and 16 strikeouts over 15 innings. He left New York's game against the Baltimore Orioles on May 19 with right forearm discomfort and three days later the Yankees announced he needed Tommy John surgery.
Typical recovery time for the ligament-replacement procedure is 12 to 18 months.
"He's on that very short list of outstanding relievers in our sport," Yankees manager Aaron Boone said after the injury. "He's been so consistent. He's such a rock for us. He's just a consistent pitcher, person, so reliable and an important part of our team and that room."
Green gets a $2.25 million salary this year, and the Blue Jays must decide after this season whether to exercise a club option calling for $9 million salaries in 2024, 2025 and 2026. Green has a conditional player option for 2024 at $6.25 million that can be exercised only if Toronto declines its option.
Green's conditional option can escalate by up to $1 million: $250,000 each for 40, 45, 50 and 55 appearances this year. If both the Blue Jays and Green decline their initial options, Toronto has a conditional option at $10.5 million annually for 2024 and 2025. Those options have $1 million in escalators for appearances this year: $500,000 each for 60 and 65 games.
The 31-year-old Green is 33-22 with a 3.17 ERA in seven major league seasons, all with the Yankees. He has 11 saves and 53 holds.
Green has struck out 494 batters and walked only 96 in 383⅔ innings. He set personal bests in 2021 with 10 wins, 67 appearances and 83⅔ innings while posting a 3.12 ERA.
To make room on the 40-man roster, left-handed pitcher Matt Gage was designated for assignment by Toronto.
The Blue Jays also appointed Jeff Ware (bullpen) and David Howell (strategy) as assistant pitching coaches. The team said the rest of last year's big league coaching staff will return this season.
The Associated Press contributed to this report.