Both Iran and Saudi Arabia have been accused of stoking the ISIS phenomenon — Iran through its support for Mr. Assad, whose crackdown has had a radicalizing effect; and Saudi Arabia through the less-accountable elements of its checkbook relationship with armed opposition groups in Syria — not to mention the ways Saudi Wahabbi doctrine has been deployed to militant ends.

But both countries now seem to be recalculating. In Tehran, there are growing concerns about the fate of Iraq, a key strategic ally, as well as ISIS’ increasingly potent threat to Mr. Assad and how far a key Iranian ally, Hezbollah, might need to be deployed in the struggle with ISIS. Iranian leaders are even reaching out to certain Syrian opposition figures.

Saudi Arabia, for its part, is clearly concerned that ISIS will breed a new generation of militants ready to turn their guns on the kingdom. Earlier this year, the Saudi authorities announced they had thwarted an ISIS plot to launch attacks across the country.