Saban said Tagovailoa's left thumb would be further evaluated by doctors in Birmingham, and he likely will be out a few days.
"Probably going to require some time for him to get back," Saban said. "I'm not going to speculate on what the injury might be."
Tagovailoa, a freshman, entered last season's national championship game in the second half when starter Jalen Hurts was benched. He threw three touchdowns against Georgia, including the game winner in overtime.
"I think in a few days Tua probably will be able to do some things, but when you hurt your thumb on your throwing hand, it's kind of hard to grip the ball. So I think that's going to be the issue, whether he is able to throw it," Saban said.
Saban added that Tagovailoa and Hurts, who split first-team reps Tuesday, will battle for the job "just like every position."
Hurts threw for 2,081 yards, 17 touchdowns and one interception last season. He also rushed for 855 yards and eight touchdowns.
Saban tried to pre-empt questions about the nature of the competition, telling reporters, "I know y'all want me to make something up that isn't there so you can make a big deal about it."
"It's two good players who can contribute to our team," Saban said, "and we're going to give them an opportunity to do that."
He added: "They're both important to our team."
Saban also reported that outside linebacker Anfernee Jennings will be held out this spring as he continues to recover from a knee injury he suffered during the Allstate Sugar Bowl.
"His rehab is going well, but we're taking it easy with this injury," Saban said. "He will be full strength by the summer."
Saban also said he was cleared by the NCAA on Tuesday to add former Tennessee coach Butch Jones to the staff in an off-field position.
Jones, who was fired by the Volunteers in December after five seasons leading the program, won't be able to coach in any hands-on capacity during practice or games.