Along with Odor, the Yankees receive cash considerations in the deal while the Rangers get minor-league outfielders Antonio Cabello and Josh Stowers in return.
The Rangers are expected to absorb the majority of the $27 million Odor has remaining on his contract, according to ESPN's Jeff Passan, who first reported a trade was in the final stages of being completed.
The 27-year-old was informed by Texas that he was not going to be a part of the club's opening day roster back on March 29, when he was designated for assignment.
Since signing a six-year, $49.5-million extension back in 2017, the second baseman has had his struggles at the plate, particularly with striking out. Odor had a career-high 178 strikeouts back in the 2019 season, which led the American League.
He had just a .167 batting average in 38 games with Texas during the shortened 2020 campaign.
The potential move to the Yankees would bring Odor face-to-face with the Toronto Blue Jays – a team he has quite the history with – on a much more regular basis.
New York, the odds on favourite to win the AL East, has started the 2021 season 2-2.