Cosart is scheduled to start Thursday's home game against the Toronto Blue Jays, which begins at 8:10 p.m. ET. The non-waiver trading deadline is at 4 p.m. ET.
Cosart, 24, is 9-7 with a 4.41 ERA in 116.1 innings this season.
Qualls, 35, is 1-2 with a 3.19 ERA and 11 saves in 15 opportunities this season. Keuchel leads the team with 10 wins and has a 2.97 ERA and 1.22 WHIP.
The Astros have the second-worst record in baseball and are in fourth place in the American League West.
- Scooby Axson