Vunipola suffered the injury on Saturday as Sarries were held to a 15-15 draw by Ospreys in the European Champions Cup.
The 25-year-old was forced off at half-time and it comes as a blow after he had just returned to the fold following a knee injury.
X-rays have confirmed the extent of the injury and he will now see a specialist, with initial rumors suggesting he will miss the next 10 weeks of action.
He was expected to be named in Eddie Jones' squad for the Six Nations on Thursday but his fitness woes should rule him out once again.
Vunipola missed last summer's British and Irish Lions tour due to a shoulder issue and Jones will now need to look at his other options at No.8.