Stoinis sustained the injury during champions Australia's 36-run defeat by India at the Oval on Sunday.
Australia, however, will wait to decide whether to bring in Marsh as a permanent tournament replacement for Stoinis - a decision that will require the approval of the World Cup's event technical committee.
International Cricket Council regulations allow injured players to be replaced during the World Cup but they cannot then rejoin their squad if they regain full fitness later on in the tournament.
"Stoinis has a bit of a side strain," Finch told reporters in Taunton on Tuesday.
"He won't be available for tomorrow. So we're going to have to juggle around especially that all-rounder spot and have got Mitch Marsh flying over just as a bit of a precaution.
"(We're) not exactly sure on how many games he'll miss," added Finch. "It's still being assessed. Over the next couple of days we'll have a clearer picture on what that looks like.
Finch, explaining how Stoinis had been injured, added: "It was towards the back end of his second spell (against India), and he came back and bowled two overs at the death.
"He was in some discomfort, it wasn't ideal, but he still managed to bowl a couple of overs."
Stoinis, 29, was out for a duck against India but took two for 62 with his lively medium pace including a superb and caught bowled to dismiss MS Dhoni.
Marsh was not included in Australia's original 15-man squad for the World Cup but is a member of the Australia A side which will begin a tour of England in the run-up to the Ashes Test series that starts after the one-day international showpiece event.
"He (Marsh) 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," said Finch."
Australia have two wins from their three World Cup matches so far, having beaten both Afghanistan and the West Indies before losing to India.
They are now fourth in the table with four points. Pakistan have three points after a win, a defeat and a no-result washout.
All 10 teams play each other once in the round-robin group stage, with the top four qualifying for the semi-finals.
New Zealand now lead the standings with six points following three wins from as many matches.