The Rays have also reportedly kept in touch with closer Craig Kimbrel.
Both pitchers have been free agents since October, with their financial demands and ties to draft-pick compensation scaring interested teams away. However, they'll no longer be subject to such compensation as of June 2.
While Keuchel would bolster their starting rotation, the Rays reportedly view their bullpen - productive thus far but relatively inexperienced - as a more significant need. Reports throughout the spring connected Tampa to Kimbrel.
Keuchel, 31, posted a 3.74 ERA over 204 2/3 innings with the Houston Astros last season. Kimbrel, 30, notched 42 saves and authored a 2.74 ERA over 62 1/3 innings with the Boston Red Sox a year ago.
Throwing big money at either pitcher would represent a huge development for the small-market Rays, who sport a major-league low $60.1-million payroll. They enter action Saturday with a 29-19 record and sit 2.5 games behind the New York Yankees for first place in the AL East.
The Yankees have also shown interest in Keuchel but have been wary about exceeding the luxury-tax threshold. They're currently $18.4 million below the tax line.
Other teams are expected to be in the mix for Keuchel and Kimbrel once June 2 arrives.