Stoinis, who has played in all three games for Australia in the tournament, picked up the injury in the match against India at The Oval on Sunday.
All-rounder Mitchell Marsh has been flown over to England as back-up, but the Australians will wait to make a call on whether he permanently replaces Stoinis in their 15-man squad.
"Marcus Stoinis has a bit of a side strain. He won't be available for tomorrow," Finch told a news conference ahead of Wednesday's match against Pakistan in Taunton.
"We've got Mitch Marsh flying over just as a bit of a precaution. He was meant to fly over with the Australia A side on Friday.
"He's just coming a couple days early as a bit of precaution, if Stoinis doesn't recover quick enough or doesn't respond to treatment well enough over the next few days."
Stoinis bowled seven overs for 62 runs against India, picking up the wickets of Virat Kohli and Mahendra Singh Dhoni.
But he was dismissed for a duck and Australia fell 36 runs short of India's total of 352 to lose the match.
Australia's pace duo Mitchell Starc and Pat Cummins found their rhythm early against India but the supporting cast of leg-spinner Adam Zampa and seamers Stoinis and Nathan Coulter-Nile struggled to keep it tight in the middle overs.
The absence of Stoinis could open up a spot for seamers Kane Richardson and Jason Behrendorff, off-spinner Nathan Lyon or middle-order batsman Shaun Marsh against Pakistan, who lost all five ODIs against Australia earlier this year.
"It's just about trying to weigh up whether we go with (Shaun) Marsh tomorrow, or we go with a extra bowler and play Alex Carey at six," Finch added.
"We've got to find a few more overs now, whether we play the extra batter or extra bowler, we'll wait and see, and especially tomorrow with the conditions."
Defending champions Australia won their opening two games against Afghanistan and West Indies.
(Reporting by Hardik Vyas in Bengaluru, editing by Ed Osmond)
