The England number eight lasted only the first half of the semi-final victory over Wasps in his latest comeback from injury but director of rugby Mark McCall insists he should be given the all-clear.
Wales full-back Liam Williams could return from the shoulder issue sustained against Gloucester on May 5 but his outlook to make the showpiece against the Premiership champions is less positive.
"Billy did a little bit of training this morning (Tuesday) and will train again on Thursday," McCall said.
"We expect him to be available to play. We've just got to be sensible with him. Hopefully he will be OK.
"Liam Williams is touch and go and we'll make a call on him on Thursday. Everyone else who played last weekend we hope to have available."
Vunipola's fitness has repercussions for England as the powerful back row has been named in Eddie Jones' squad bound for South Africa next month.
The 25-year-old returned from a fractured arm against London Irish earlier this month only to be struck by the hamstring complaint that has limited his game time ahead of the three-Test series against the Springboks.
"If his hamstring is OK and everything is OK post final, then it would be a good thing for him to go on the tour," McCall said.
"Clearly if he's carrying something then it wouldn't be the best for him to go on a three-week tour."