The Los Angeles Dodgers are among the teams trying to land the veteran right-hander, with Morosi adding the Indians are interested in Dodgers outfielder Alex Verdugo in a potential package for Kluber.
Kluber finished third in American League Cy Young Award voting last season, winning a career-high 20 games to go with a 2.89 ERA and 222 strikeouts in 215 innings.
The 32-year-old has also been one of the most consistent pitchers in baseball during a dominant five-year stretch. From 2014 to 2018, Kluber won two Cy Young Awards and finished in the top three in voting four times. He was ninth during his lone "down" year in 2015.
He also had an 83-45 record with a 2.85 ERA during that stretch, topping 200 strikeouts and 200 innings in each season.
And he was dominant during Cleveland's run to the World Series in 2016, going 4-1 with a 1.83 ERA in 34.1 innings.
It's clear the impact he could have, which has made him a popular target in trade talks. Morosi reported in November the Dodgers and Indians discussed a swap that included Yasiel Puig. Last week he noted the Cincinnati Reds were open to a deal.
Los Angeles might make the most sense as a contender that could use more frontline pitching depth to help it get over the top, though anyone would love Kluber on their staff.
The veteran is set to make $17 million in 2019 with team options for 2020 and 2021, per Spotrac, so a move would also provide flexibility for whichever team wins the sweepstakes.