Navaira had been listed in critical condition for more than three weeks with injuries suffered in a tour bus accident Easter Sunday.
A hospital official said Navaira "reached a significant milestone in his recovery," the San Antonio Express News reported Thursday.
Hospital representative Alex Rodriguez said the upgrade is important, but stressed Navaira has a long way to go before being released.
"What the fact that he has been upgraded to fair means is that he's recovered significantly, but he is not walking out of the hospital," Rodriguez said. "We have three conditions we report -- critical, fair and good. He is right in the middle. Neurologically he is better; his lung is better."
Since the accident, in which he was hurled through the windshield of his tour bus, Navaira has had brain surgery twice to remove a blood clot and relieve pressure from swelling. He also underwent another procedure to repair an arterial wall in his right lung.