Greene, 30, will not report with the rest of the team's position players Tuesday and could miss significant time, according to major-league sources.
Rangers general manager Jon Daniels declined comment when contacted Sunday night, saying the team planned to address the issue on Monday.
The Rangers signed Greene to a one-year, $500,000 free-agent contract on Jan. 11, intending for him to replace Omar Vizquel as their utility infielder. Vizquel had signed a one-year, $1.375 million free-agent deal with the White Sox earlier in the offseason.
Felipe Lopez is the most prominent infielder remaining on the free-agent market. The Rangers also could consider internal candidates or pursue other options later in spring training, when utility infielders are readily available.
Greene twice went on the disabled list last season with social anxiety disorder, appearing in only 77 games for the St. Louis Cardinals.