The Royals indicated Eibner would have an MRI on Wednesday.
He was pursuing a ball hit in the left-center gap by the Rays' Brad Miller when he stepped awkwardly and immediately grabbed at the leg, tossing the ball to center fielder Lorenzo Cain to relay it to the infield.
Royals trainer Nick Kenney rushed out to examine Eibner.
"We hope it's a sprained ankle," Royals manager Ned Yost said.
But Yost indicated it certainly looked like a disabled-list move.
"Pretty much," Yost said. "We'll look at it tomorrow."
Yost said he didn't have a good view of the play.
"I didn't really see it, couldn't tell what happened," he said. "I saw him go down, and I was, 'OK, get back up.' But he tossed the ball to Lorenzo and didn't get back up. Nick said his foot just gave out."
The Royals starters all huddled around Eibner as they waited for the cart to take him off the field.
"Any time you see someone go down like that, you get extremely worried," first baseman Eric Hosmer said. "But judging by his body language, maybe it's not that bad. We'll know when they get pictures of it."
Eibner is 6-for-13 since his callup, with three doubles.
Jeffrey Flanagan has covered the Royals since 1991, and for MLB.com since 2015. Follow him on Twitter @FlannyMLB. This story was not subject to the approval of Major League Baseball or its clubs.