Kluber, 32, has spent all eight seasons in the majors with the Indians. He has been at the center of trade rumors since the end of the season, with the Los Angeles Dodgers, Milwaukee Brewers and Cincinnati Reds rumored to be the most interested.
The Reds have since backed off, according to multiple reports.
"While a Kluber trade remains possible, sources say the chances are diminishing by the day," MLB.com reported. "It is now more likely than not that Kluber will remain with Cleveland as the new season begins."
Kluber was credited with 33 starts last season and finished with a 20-7 record and a 2.89 ERA.
The Indians finished 91-71 to win the AL Central Division, but were swept by the Houston Astros in three games in the first round of the playoffs.
"The Indians entered the offseason with an eye toward reducing their payroll, and they have done so without trading Kluber," MLB.com reported. "The Indians are on pace to begin the season with a payroll of around $115 million, a reduction of roughly $20 million from Opening Day 2018, according to Cot's Baseball Contracts."