Masterson reportedly enjoys playing for manager Terry Francona, was encouraged by the team's playoff berth last season and has family in Ohio.
The Indians locked up outfielder Michael Brantley to a four-year extension with an option for a fifth-year this offseason. Pitcher Ubaldo Jimenez chose to go to Baltimore for a long-term contract after declining the Indians' one-year, $14 million qualifying offer.
Masterson was an All-Star last season, finishing the year with a 3.45 ERA and 195 strikeouts in 193 innings pitched.
Securing the 28-year-old, who has been a lynchpin of the pitching staff, would be huge for long-term stability atop the rotation.