The 25-year-old, who returned to action this month after undergoing knee surgery in September, played only one test for England last year after a succession of injury problems and was ruled out of the British and Irish Lions tour of New Zealand.
"Billy is due to see a specialist later this week, when a time-frame for recovery may be defined," Saracens said in a statement on Sunday.
"Saracens will issue a further statement once Billy has seen a specialist."
England coach Eddie Jones names his Six Nations squad on Thursday and the defending champions' first match is against Italy on Feb. 4.
Nathan Hughes, Vunipola's usual deputy, has been ruled out of the start of the tournament with a knee injury and Jones could call up 23-year-old Exeter number eight Sam Simmonds.
(Reporting by Ed Osmond; Editing by Clare Fallon)
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