The deal is worth $8 million guaranteed and could be worth $13 million if he reaches incentives, the source said, confirming multiple reports.
Kluber, who turns 36 in April, was limited to 80 innings last season with the New York Yankees because of a shoulder strain that kept him out for more than three months. He went 5-3 with a 3.83 ERA in 16 starts for the Yankees and pitched a no-hitter in May, striking out nine batters in a 2-0 victory over the Texas Rangers.
The two-time Cy Young Award winner hasn't pitched a full season since 2018. He spent his first nine seasons with Cleveland and was a three-time All-Star, winning 98 games with the franchise and striking out more than 200 batters five times.
For his career, Kluber is 103-61 with a 3.19 ERA in 1,422⅔ innings over 11 seasons for Cleveland, Texas and the Yankees.
ESPN's Jesse Rogers contributed to this report.