"It's a little bit uncertain, Billy is in a race against time to be ready for the quarter-final," McCall told reporters.
"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."
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
The arm injury was the latest setback in a wretched spell for Vunipola, who also missed last year's British and Lions Tour and England's autumn internationals due to serious shoulder and knee injuries respectively.
Vunipola last played for England in their defeat by Ireland in the final match of the 2017 Six Nations campaign and head coach Eddie Jones will be keen to see the forward return for the three-test tour of South Africa in June.
(Reporting by Aditi Prakash in Bengaluru)
