The San Diego Padres have been in contact with the Indians about a possible Kluber trade since the new year began, sources told MLB.com's Jon Morosi, who reports a deal wasn't close as of Friday night.
The Los Angeles Dodgers have also been involved in separate negotiations with Cleveland about the two-time Cy Young winner, a source told Morosi.
San Diego is trying to acquire Kluber without giving up any of their top five prospects, Morosi's sources tell him, with the Indians having interest in left-hander Adrian Morejon.
Morejon is the organization's No. 6 prospect, per MLB Pipeline.
Morosi doesn't mention names in the Dodgers' system, but Cleveland was reportedly interested in outfield prospect Alex Verdugo in December.
Both the Dodgers and Padres have been pursuing Kluber since last month. The Milwaukee Brewers and Philadelphia Phillies have also shown interest.
Kluber, 32, is eligible for free agency following next season but has two team options attached to his contract for 2020 and 2021.
He's set to earn $17 million in 2019 after posting a 20-7 record with a 2.89 ERA across 215 innings last season.