The 28-year-old quick, a 2010-11 and 2013-14 Ashes tourist, missed the opening tour match in Perth after sustaining the injury in practise last week.
"...(Finn) will now return to the UK in the 48 hours where he will meet a knee specialist to ascertain whether he will have an operation," the ECB said in a statement, adding the board will announce a replacement later.
All-rounder Moeen Ali, who has a side strain, will miss this week's day-night match against a Cricket Australia XI but was on track to be available for a return to action in the warm-up game in Towsnville starting Nov. 15.
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"Mo Ali ... more precautionary now, we will just leave him out of this one and make sure he's right," England coach Trevor Bayliss told reporters in Adelaide.
Finn was added to the squad as a replacement for Ben Stokes after the all-rounder was suspended following his arrest on suspicion of causing actual bodily harm outside a nightclub.
Bayliss also responded to reports in the Australian media on Tuesday that said Stokes would soon face a England and Wales Cricket Board disciplinary hearing and could join the squad in time for the third test.
"I haven't spoken (to anyone) or heard of anything since we have been here," said Bayliss.
"If we concentrate on that, and wondering if he is coming or not, that might take the focus off what we're trying to achieve so we have just been going about preparing with the players we have got here.
"If he happens to turn up at some stage, it's a bonus. If he doesn't, bad luck, we will just get on with what we have got."
ECB director of cricket Andrew Strauss said England were looking for "clarity" over Stokes's position.
England begin their campaign to retain the Ashes at the Gabba in Brisbane on Nov. 23.
(Reporting by Nick Mulvenney and Hardik Vyas; editing by Amlan Chakraborty)
