A's manager Bob Melvin said Burns was expected to join the team before Monday night's game in Houston.
Gentry was injured Sunday at Texas when he hit by a pitch. He was on the disabled list at the start of the season with a lower back strain.
Gentry was hitting .264 with 10 RBIs in 80 games with Oakland. Burns was batting .250 with a home run and 23 RBIs in 91 games with Midland.