The British & Irish Lion sustained the injury trying to stop South Africa's Jean de Villiers scoring his team's first try in Saturday's 24-15 win over Wales in Cardiff.
"If he needs an operation it could be five months," Wales coach Warren Gatland told the BBC.
"He's pretty disappointed. The physio said it might settle and he could be back in December. We will wait and see what happens."
Gatland was already without Jamie Roberts, who usually partners Davies in the centre, and winger Alex Cuthbert for the November tests while prop Adam Jones suffered a calf injury during the game against South Africa.
Wales play Argentina on Saturday followed by Tonga and Australia.
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