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Tolleson began 2016 as Texas' closer but eventually was replaced after he struggled to a 7.38 ERA and 1.73 WHIP over 37 appearances. He eventually landed on the 60-day disabled list in August with a lower back strain.
Tolleson has proven in the past that he can be a workhorse in the bullpen when healthy.
The right-hander led the Rangers with 174 appearances over his three seasons with the club, while also collecting 46 saves in that span. Tolleson also recorded two scoreless appearances for Texas during the 2015 American League Division Series against Toronto.
If he does sign with Tampa Bay, Tolleson would bring a veteran presence to a bullpen that also includes Brad Boxberger, Xavier Cedeno and Alex Colome. Rays relievers finished with a 4.09 ERA in 2016, the fifth-highest mark in the American League.
Matt Kelly is a reporter for MLB.com based in New York. Follow him on Twitter at @mattkellyMLB. This story was not subject to the approval of Major League Baseball or its clubs.