The numbers are flying as to how many countries have signed on to the United States-led coalition to fight the militant group Islamic State that’s in control of parts of Iraq and Syria. Obama administration officials say more than three dozen regional and Western states have signed on.

But there are overt participants, and then there are the hidden partners – some of whom will be more central to the anti-Islamic State (IS) effort than, say, Albania, which has agreed to provide some arms and ammunition to Iraqi Kurdish peshmerga forces.

Among these “hidden partners” that won’t publicly enlist in the coalition but will very likely play crucial roles in the fight against IS, also known as ISIS or ISIL: Russia and Iran. And some Arab countries will help but will prefer to keep quiet about their assistance.

As the battle already under way in Iraq shifts to Syria, the US may even find itself teaming up with Jabhat al-Nusra, some Middle East diplomats and analysts speculate. The Nusra Front rebel group is one of IS’s stronger foes in northern Syria at the moment – but it comes with the inconvenient qualification of being an Al Qaeda affiliate.

And then there’s the case of Syrian President Bashar al-Assad, whose regime is rejecting any aerial targeting of IS on its territory without its permission – but which privately views the anticipated airstrikes as a potential lifeline, some regional analysts say.

“Assad will benefit [from US action against IS inside Syria] but not because the [Obama] administration wants him to,” says James Phillips, Middle East expert at the Heritage Foundation in Washington. “That’s just one example of how US action could benefit those who ultimately are working at cross-purposes with what we are trying to do,” he adds, “but I don’t think that means we see them as part of a tacit or hidden coalition.”

Speculation about off-the-books alliances to further the anti-IS fight has focused most intensely on Iran.

Secretary of State John Kerry fed the frenzy Monday by repeating the US position that it is willing to discuss the IS threat with the Iranians, although he ruled out any direct coordination of military efforts.

Speaking after an international conference in Paris Monday on addressing the IS threat in Iraq, Secretary Kerry said the US had rejected Shiite Iran’s participation in the conference because several Sunni Arab countries that are key to pushing back IS in Iraq – including Saudi Arabia – had flatly refused to take part if Iran were present. But the US does not oppose addressing the issue with Iran in other venues, he added, including on the margins of ongoing talks on Iran’s nuclear program.

Calling an open “channel of communication ... common sense,” Kerry said the US is not “opposed to the idea of communicating to find out if they will come on board or ... whether there is the possibility of a change” in Iran’s actions in the region. Iran supports Shiite militias in Iraq that are known to have brutalized Sunni villages, and along with Russia is a crucial underwriter of Mr. Assad.

Russia, which did take part in the Paris conference, is another important regional player that won’t sign on to President Obama’s coalition but could end up assisting it.

Last week Kerry went out of his way to suggest that Russia has a common interest with the US and its partners in seeing that IS is defeated. He said that, just as the US and Western Europe are worried about the thousands of “foreign fighters” who are aiding IS but could eventually return home to commit terrorist acts, Russia, too, could face local terrorists “who will take an example from ISIL.”

At the Paris conference, Russian Foreign Minister Sergey Lavrov sought to outflank Kerry and the US on coalition building, insisting that both Syria and Iran should be viewed as “natural allies” in the effort against IS. “The extremists,” he said, will “use any disagreements in our positions to tear apart the united front of states acting against them.”

Russia might conceivably be seen as a potential partner in Iraq, says Mr. Phillips of Heritage, for example in helping to supply the Iraqi security forces. But, he says, the US should avoid falling prey to the idea that either Russia or Iran could in any way be useful in the fight against IS in Syria.

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“Sometimes the enemy of my enemy is still my enemy,” he says, noting that no matter how much Russia and Iran may favor the defeat of IS, they are still the biggest backers of Assad.

“And I don’t think ISIS can be defeated as long as Assad is there,” Phillips adds, saying that Assad has allowed IS to flourish “as a useful foil to advance his narrative that the opposition is all terrorists.”