Davidson will be used as an infielder and could pitch as well, giving the Rangers their own two-way weapon.
Davidson, 27, got off to a hot start with the White Sox in 2018, hitting three home runs on Opening Day and nine total in the first month. But he finished the season with a disappointing 20 long balls to go with just a .228 batting average and 165 strikeouts.
He played the majority of his games last season at designated hitter. The White Sox chose not to tender him a contract, making him a free agent. He also saw time at first base and third.
Davidson has been working pitching into his offseason repertoire, after three scoreless relief appearances last season for Chicago.
"Who knows what could happen?" Davidson said after one of those outings in 2018. "The game is changing a little bit. In the next couple of years there might be more two-way players. You're seeing guys getting drafted who are two-way guys. Teams are a little more open to it."
Davidson was drafted by Arizona in the first round in 2009, then traded to the White Sox in December 2013 for reliever Addison Reed.