Dellucci, 35, batted .275 for the Indians in 40 at-bats earlier this year.
He cleared waivers on June 2, and can sign with any team for the prorated minimum salary. The Rangers recently lost left-handed hitting slugger Josh Hamilton to an abdominal injury that might knock him out for much of the rest of the season. If they were to sign him, Dellucci would likely receive a Triple-A deal to start.
Dellucci, who has played in 1,091 games and slugged 101 homers in a 13-year career, has also drawn interest from the Toronto Blue Jays.
Buster Olney is a senior writer for ESPN The Magazine.