The 23-year-old requires surgery after picking up a serious shoulder injury during the first half of his side's 27-20 Heineken Cup loss to Glasgow last weekend.
Clark was forced from the pitch after 37 minutes and subsequent tests have now shown he will require an operation to repair damage to his AC joint.
The former Leeds Carnegie forward had been called up to England's Elite Performance Squad by Stuart Lancaster for the Six Nations, but now looks unlikely to play any part.
Clark tweeted of his injury on Tuesday afternoon: "My shoulder needs an operation so I will be out of action for the foreseeable future. That's the game."
The Rugby Football Union subsequently announced that Clark will undergo surgery on Wednesday.
England have already called up Worcester's Matt Kvesic and Saracens lock Moritz Botha as forward cover after injuries to both Clark and Tom Johnson (knee) during last weekend's European fixtures.
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