Best started three pool games against the USA, Australia and Italy, impressing at set piece time and in the loose and scoring his first Test try in over two years.
Commenting on the hooker's condition, Ireland team manager Paul McNaughton said: "Rory's doubtful for the weekend but we haven't written him off just yet. We'll assess it tomorrow (Tuesday) 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."
If the injury is sufficient enough to rule Best out of the tournament, he will be the second Irish hooker to have his World Cup prematurely ended after a calf problem led to Jerry Flannery returning home.
Sean Cronin is standing by to take Best's place in the starting line-up. He had an 80-minute runout against Russia and replaced Best in the second half of the Italy game. Recent call-up Damien Varley is next in line.
McNaughton added: "Rory has been playing fantastic rugby and it would be a blow if he's not available. 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."
The squad and management have transferred from Dunedin to Wellington, settling into their base this week ahead of Ireland's first Rugby World Cup quarter-final since 2003.
The team and replacements to face Wales in next Saturday's showdown at Wellington Regional Stadium will be announced on Wednesday (1.15pm New Zealand time/1.15am Irish time).
Follow the Ireland team in New Zealand on www.twitter.com/irfurugby.