Montero, 22, was taken out of Wednesday's game after getting hit in the mask by a foul ball from Robert Andino in the fifth inning.
Manager Eric Wedge said Montero will be evaluated daily until he's cleared, but wouldn't need to go on the seven-day DL because of the upcoming All-Star break after this weekend.
"He's closer," Wedge said of Montero. "I know he's frustrated and wants to play, but you can't mess around with those things. We'll give him the test every day and see how he reacts to it and go from there. We'll see."
Montero indicated his own situation wasn't that serious.
"He hit me pretty good, but it's not a big deal," Montero said. "I don't know what they're going to do, but I'm ready. It's whatever they decide, but I'm fine."
Franklin Gutierrez remains on disabled list with a more-severe concussion suffered June 29. Wedge said he likely won't be ready even when his second seven-day stint is up on July 13.
Gutierrez was hit in the head by a pickoff throw by the Red Sox's Franklin Morales.
"He's better, but he's not there yet," Wedge said. "He still has bouts of dizziness and the affects of it. The first day he's eligible to come off would be Friday after the break, but that may be a little optimistic."
"Guti's was worse because he didn't have any protection," said Montero. "Thank God I have the mask."