The 27-year-old Odor did not make the Rangers' Opening Day roster and was designated for assignment on April 1. He is a left-handed hitter, and the Yankees have a mostly right-handed lineup.
"Hopefully we can help him get that talent out that we've seen flash over the years,'' Yankees manager Aaron Boone said.
Odor is a second baseman but also played third in spring training.
Texas is expected to absorb almost the entirety of the $27 million owed to Odor, sources told ESPN's Jeff Passan. He's due $12 million this year, $12 million in 2022 and a $3 million buyout on a $13 million club option in 2023.
Odor has a .237 career average with 146 homers and 458 RBIs in seven major league seasons. He hit .167 with 10 homers and 30 RBIs in the pandemic-shortened 2020 season and batted .200 with two homers and five RBIs in 35 spring training at-bats for the Rangers.
New York starts right-handed hitters with the exceptions of switch-hitting center fielder Aaron Hicks and left-handed-hitting first baseman Jay Bruce, who is starting while Luke Voit recovers from knee surgery.
New York optioned right-hander Michael King to its alternate training site to open a space on the active roster. King pitched six innings of scoreless, one-hit relief against Toronto on Sunday.
Infielder Thairo Estrada was designated for assignment to open a spot on the 40-man roster.
The 20-year-old Cabello hit .251 with 19 doubles, nine triples, eight homers and 40 RBIs in the minors from 2018 to 2019.
Stowers, 24, has hit .268 with 39 doubles, 12 homers and 68 RBIs over two minor league seasons.
The Associated Press contributed to this report.