Verrett was selected out of Baylor in the third round of the 2011 amateur draft by the New York Mets.
To make room on the 40-man roster, Texas put outfielder Antoan Richardson on the 60-day disabled list. He had surgery last week to repair a herniated disk in his lower back.
Baltimore selected Verrett from the Mets in the winter-meeting draft during the offseason. He had a 1.93 ERA in nine spring training games for the Orioles.
The 24-year-old Verrett was 11-5 with a 4.33 ERA in 28 starts for Triple-A Las Vegas last season, and had a 28-13 record with a 3.90 ERA in 69 starts in the Mets system.