Saracens take on Leinster on March 31 and director of rugby Mark McCall says Vunipola is a major doubt for the clash.
The England number eight has been missing since January with a fractured arm, missing the Six Nations as a result, and McCall says he has no intention of taking any risks with the 25-year-old.
"It's a little bit uncertain, Billy is in a race against time to be ready for the quarter-final," said McCall, as reported by the Evening Standard.
"We'll make a call on what the surgeon says. If the arm isn't right to play then we won't risk him. He's been injured enough, we don't want to re-injure him.
"Billy is very close to coming back but whether he makes the quarter-final we're not sure yet."