Best sprained the AC join in his right shoulder during yesterday's 36-6 victory over Italy at Otago Stadium and is receiving treatment.
A replacement for the Ulster hooker, who has been playing the best rugby of his Ireland career, has yet to be summoned and a clearer picture over his condition will emerge on Tuesday.
"Rory's doubtful for the weekend but we haven't written him off just yet," said manager Paul McNaughton.
"We'll assess it tomorrow and have more information then. He's definitely doubtful.
"He's getting manipulation at the moment. We'll have a better idea tomorrow after a morning's treatment."
Ireland have already lost one hooker during the World Cup after Jerry Flannery returned home two and a half weeks ago with a calf problem.
"Rory has been playing fantastic rugby and it would be a blow if he's not available," added McNaughton.
"But we have two hookers out here and back up in Ireland if we need it.
"Other teams have had injury problems and we must get on with it."
McNaughton revealed prop Cian Healy was interviewed about the attempted eye gouge by Italy hooker Leonardo Ghiraldini during the first half of yesterday's critical Pool C showdown.
The citing commissioner has yet to cite Ghiraldini for the incident that provoked a furious response from Healy.
"We were asked questions and asked to make a brief statement on it to the match commissioner. We did that in the ground," said McNaughton.
"Cian answered several questions put to him by the match commissioner and gave a statement, but we haven't heard anything since."
Ireland arrived in Wellington today ahead of the weekend's mouthwatering showdown with Wales having spent the previous seven nights in Dunedin.
Backs coach Alan Gaffney has been impressed by Wales' form in New Zealand and believes they have profited from a change in approach.
"We've seen all of Wales' games during the World Cup and the Six Nations. They're playing well and could have beaten South Africa," he said.
"They've changed their style, which had become predictable. People said it was attractive, but it was predictable.
"Full credit to (head coach) Warren Gatland and his coaches. They've changed their attack, which is now more potent."
Ireland would have exited the World Cup had they lost to Italy and Gaffney admits anxiety levels were running high.
"I can't recall when I've been as nervous as that before a game because it was so huge," he said.
"We knew the pressure was on and it would have been a travesty after the way we played against Australia to have missed out on the quarters."
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