The Middlesex quick suffered the left hamstring tear just two balls into his second-innings spell against Pakistan in the fourth Test at The Oval on Sunday and has been let out of England's squad for the upcoming five-match one-day international series.
Finn, 27, is not expected to play international cricket again this season.
But the 6ft 7in veteran of 35 Tests and 65 ODIs, is aiming to be involved in County Championship action in September.
"He is anticipated to be available in approximately four weeks.
"The 27-year-old will start a rehabilitation programme with Middlesex and England, and is aiming to play a part in Middlesex's Championship campaign in September.
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