Verrett had been a member of the Mets organization from the 2011 draft through December's winter meetings, when he was selected by the Baltimore Orioles in the Rule 5 draft.
Verrett was claimed off Rule 5 waivers by Texas this week after Baltimore began the process of offering Verrett back to the Mets.
Barring something unforeseen, Verrett will make the Rangers' Opening Day roster, general manager Jon Daniels told local media Friday.
Verrett, 24, will need to stick on the major league roster of the Rangers for the entire season to officially become Texas' property.
He went 11-5 with a 4.33 ERA in 28 starts for Triple-A Las Vegas last season.