Jackson, the 45th overall pick in the 2010 draft, is looking to make his major league debut. He was 2-2 with a 4.05 ERA in 33 Triple-A games.
Bass worked 51 2-3 innings in 26 relief appearances at Texas before being optioned July 27.
Claudio didn't appear in a game after his recall July 29 for his third stint with Texas this year. Klein has a 6.75 ERA in four stints.
Left-hander Wandy Rodriguez became a free agent Thursday after clearing release waivers.