Finn was injured in practice last week in Perth shortly after the squad arrived in Australia and had already been ruled out of England's first two tour matches.
Replacement bowler Tom Curran, who has played Twenty20 and One-day International cricket for England, was preparing to leave for Australia on Wednesday.
A tall fast bowler, Finn has taken 125 wickets in 36 tests but has played only two of those matches in Australia, both in 2010.
The Ashes series starts Nov. 23 at the Gabba in Brisbane. England has two more warmup games, including a four-day pink ball match in Adelaide starting Wednesday, to prepare for its Ashes defense.
England coach Trevor Bayliss said selectors were still considering their options on a batting a spot and another pace bowling option ahead of the first Test.
Allrounder Moeen Ali has a side strain but is expected to return for the last of the warmup matches next week.