Bassitt was "conscious and aware," the A's said in a tweet, adding that he was being taken to a local hospital.
Oakland manager Bob Melvin said after the game, a 9-0 A's loss, that Bassitt never lost consciousness.
"We don't think the eye is a problem at this point. It felt like it was below it. He's got some cuts that [need] some stitches," said Melvin, who took no questions. "He's in a scan and we'll know more about potential fractures tomorrow."
Bassitt, 32, immediately fell to the ground after being struck by the ball, which came off White Sox center fielder Brian Goodwin's bat at 100.1 mph.
Teammates as well as the A's training staff rushed to his aid at the mound, where Bassitt remained down for several minutes. He was helped onto a golf cart and taken off the field.
Bassitt, who is 12-4 with a 3.22 ERA this season, was replaced by reliever Burch Smith.
Goodwin was not made available to the media after the game, but he expressed support for Bassitt in a tweet later Tuesday.
Most definitely, Chris, you are in my prayers for sure brother 🙏🏾 https://t.co/tR6478Tb43— B. Goooood 🏁 (@thatBoy_GOOD1) August 18, 2021
Chicago manager Tony La Russa said Bassitt was "the No. 1 concern."
"We're all really hoping that he escaped with just a bruise," La Russa said. "There was a dark cloud around that whole game."
A's players also did not speak to the media.
"It's a lot bigger than baseball at that point," White Sox third baseman Jake Lamb said. "Anytime you get hit in the head like that, it's a very scary situation."
White Sox pitcher Reynaldo Lopez also said he was shaken by what happened.
"When you see something like that, it's hard to digest because ... as a pitcher, we know that that can happen," he said through a translator. "It's like, that could happen to me and it's scary. Every time you see something like that it is tough to swallow.''