The barnstorming number eight missed the British and Irish Lions tour due to the injury, having returned from a four-month layoff due to a knee problem in March.
Sarries director of rugby Mark McCall confirmed the England international is not scheduled to return until October, although he did not rule out Vunipola playing before then.
"He's in great condition at the moment and chomping at the bit," McCall said.
"Billy's due back some time in October, but in true Billy style he'll make it back considerably before that.
"I'll be told by the medics when we can pick him but it won't be long."
Sarries had six players representing the Lions in New Zealand and their season opener with Northampton Saints on September 2 may come too soon for the likes of Jamie George, Owen Farrell, Maro Itoje and Liam Williams, who could have to wait for his debut after joining from Scarlets.
But McCall assured he will judge each player on a case-by-case basis.
"We've got a game coming up in 10 days and we need to take a long view with most of them," said McCall.
"To rush them back too early if we didn't have to would be a mistake in a couple of cases.
"But where the team needs that to happen then it may be that we need to bring one or two back for the first game.
"You need to look back over the past couple of seasons and who played how many minutes.
"A couple of players didn't play that much last year. Someone like George Kruis didn't play in the last two Lions Test matches, others missed a bit of rugby.
"With Mako [Vunipola] and Owen we have some tough choices to make.
"As a club we're not looking to get our pound of flesh. We're looking at the welfare of the player.
"I'm trying to make good choices based on good reasons as to when they come back."
Photo Getty Images. Caption: England and Saracens star Billy Vunipola