Kluber, 33, posted a 5.80 ERA in seven starts last season. He missed most of the season with a broken ulna bone in his right forearm and an oblique strain.
The three-time All-Star spent nine years with the Indians, posting a 3.16 ERA in 203 starts. He won the American League Cy Young Award twice — in 2014 and 2017.
Kluber will make $17.5 million in 2020. He has an $18 million club option in 2021 that becomes guaranteed if he pitches 160 or more innings next season and doesn't finish the year on the injured list.
Ken Rosenthal of The Athletic says the Indians will receive outfielder Delino DeShields Jr. and right-hander Emmanuel Clase in the deal.
Deshields, 27, has posted a .246/.326/.347 slash line in five seasons with the Rangers. He's played in at least 106 games in four of his five seasons.
Clase posted a 2.31 ERA in 21 appearances for the Rangers last season. He had 21 strikeouts and allowed just six earned runs in 23⅓ innings.
The 21-year-old relies on two pitches — a cutter [fastball] and a slider. His fastball routinely hits 99, but has topped out at 102 mph.
Clase is certainly the center piece of this deal for the Tribe. He brings an element to the bullpen that the Indians lacked last season — a hard throwing right-hander that can take some pressure off of closer Brad Hand.