The 21-year-old suffered a suspected broken collarbone in Saturday's 2-0 win over Motherwell, following a challenge from Keith Lasley.
Celtic confirmed on the club's official website that the Wales international will have a consultation on the problem.
"Adam Matthews will see a specialist on Monday," a Celtic statement read.
"However, there is the possibility of surgery and Adam could be out for a couple of months."
If Matthews does miss the next two months, it would mean he will not be involved for the remainder of Celtic's Champions League Group H campaign.
The full-back, who has also been deployed in a midfield role this season, will definitely miss his country's two remaining Group A World Cup qualifiers, against Macedonia on Friday and Belgium the following Tuesday.