Vunipola went off injured in Saracens' 13-3 Champions Cup victory over Glasgow on Sunday, a match in which his older brother Mako and Nick Isiekwe also picked up injuries.
The injury ruled him out of England's squad for their November internationals.
Sinckler cited after bad-tempered Lions defeat to South Africa
"Billy has fractured his left arm and will have surgery on Tuesday. He is expected to be out of action for around 12 weeks," Saracens said on their website https://www.saracens.com/news-article/injury-update-nick-isiekwe-billy-vunipola-mako-vunipola.
Prop Mako suffered a calf strain while forward Isiekwe added to the national team's injury woes by suffering an ankle sprain. The fitness of both players will be monitored.
The trio were part of England's training squad for the Autumn series against South Africa, New Zealand, Japan and Australia next month.
England have been unlucky with injuries, with lock Joe Launchbury ruled out after knee surgery last week while Bath fullback Anthony Watson and Exeter forward Sam Simmonds will also miss next month's games.
England coach Eddie Jones names his squad for the November games on Thursday.
South Africa secure second Test 27-9 victory over the Lions at Cape Town Stadium