The former Cardiff City full-back, 21, was taken off on a stretcher after 13 minutes of the Bhoys' 2-0 victory over Motherwell on Saturday.
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A statement on Celtic's website revealed that Matthews will go under the knife on Wednesday and added: "Everyone at Celtic sends their best wishes to Adam ahead of the surgery and for the weeks ahead."
The defender, who can also play on the wing, has made 91 appearances for the Scottish champions since his move from Cardiff in 2011.
His injury adds to a lengthy list of absences for Wales boss Chris Coleman ahead of their final World Cup qualifiers against Macedonia and Belgium.
Nine players have already withdrawn ahead of the two fixtures that the Welsh must win to avoid finishing bottom of Group A - which could have damaging repercussions on their seeding in qualifying for Euro 2016.