(CNN) A U.S. proposal to deepen military cooperation with Russia in Syria has sparked a rift at the highest levels of the Obama administration, with the Pentagon openly challenging an idea that the top U.S. diplomat calls critical to moving Syria forward.

Even as Secretary of State John Kerry is in Moscow negotiating a possible agreement that could lead to the U.S. sharing classified intelligence with Russia over military strikes both countries are conducting in Syria, the Pentagon is making it clear that Defense Secretary Ash Carter remains skeptical of Russian military activities in the war-torn country.

Both the U.S. and Russia have aircraft that conduct operations over Syria, and the new agreement, a copy of which was obtained by CNN, could allow for targeting and carrying out joint air strikes against ISIS and al-Nusra, the al-Qaeda branch in Syria.

"The secretary of defense has been clear that he has been skeptical of Russia's activities in Syria and we have reason for that. There's plenty of reasons for that skepticism," Pentagon Press Secretary Peter Cook told reporters Thursday. "And I think he maintains that skepticism."

The Pentagon is strongly signaling that whenever agreements are reached, the Defense Department wants to see guarantees Moscow will live up to its part of the bargain.

"If Russia can behave in a more positive fashion, then we would be open to that conversation," Cook said.

But Kerry is pressing ahead.

A senior administration official told CNN that Brett McGurk and Rob Malley, two key administration aides who deal with the ISIS file, were in Zurich this week negotiating the modalities of the plan with Lavrov and other top Russian officials.

But they didn't make much progress, officials said, and Kerry is now going to Moscow to see if he can push the negotiations further.

"That proposal will be discussed" in series of meetings Kerry is having in Moscow, including a session with Russian Foreign Minister Sergey Lavrov, a senior U.S. official told CNN.

The US sees the major trade off as deeper cooperation with Russia, which they want, in exchange for Russia making sure that the Syrian airforce is grounded.

"The Russians have been asking for this for some time but haven't put a quid on the table. It wasn't clear what they would do in exchange. If they can ground Assad's air force and make him do something on the political track, that would be significant," the official said.

The official noted that in told NBC in an interview aired Thursday that President Putin never asked him to ground his Air Force

Kerry has described the administration's patience with Russia as "limited," and senior officials say that impatience is growing by the day. One official described the effort to negotiate with the Russians as a "Hail Mary" to test Russia's seriousness.

The idea of sharing classified military intelligence information with Moscow has set off a firestorm of controversy between the State Department and National Security Council, which appear to favor the idea, and the Pentagon and intelligence community, which are deeply opposed.

The White House is signaling its own support U.S.-Russian military cooperation in Syria is conditional.

"I know there are some speculation that an agreement may be reached to do so. But it's not clear that will happen," White House Press Secretary Josh Earnest told reporters Thursday.

The proposal being discussed would potentially set up a Joint Operation Center for the two sides to work out of, according to a draft first disclosed by the Washington Post.

The two countries would work together to define where al-Nusra is located in Syria and each country's military would submit to the other a list of "actionable targets" pertaining to al-Nusra at least 24 hours before a strike, according to the draft. The strikes would be confined to the designated jointly agreed upon Nusra areas.

"Only those targets that both participants agree are actionable will be further developed for strikes," the document reads. It would also involve delineating the location of "moderate opposition forces."

The arrangement has troubled Defense Secretary Ash Carter, who has largely opposed the proposal, and defense sources have raised questions about the benefits that would accrue to Russia under the deal and whether Moscow would keep its commitments.

One official characterized the proposal as "the Russians getting everything they want -- U.S. cooperation -- and the U.S. getting nothing in return."

As reported previously by CNN, the accord call for grounding the aircraft of Syrian President Bashar al Assad except for medical, humanitarian, and rescue missions, thereby possibly putting an end to the regime's use of barrel bombs against opposition controlled areas.

Though that's a major U.S. interest, it's acknowledged that Syrian ground units would still go after the moderate opposition as it continues to battle them in a years-long civil war and even potentially civilians.

Better targeting al-Nusra is also a high U.S. priority. The group has gained strength recently and is considered one of the more capable groups fighting the Assad regime in some northeastern parts of Syria.

The challenge in northeastern Syria is that the U.S. has been unable to strike al-Nusra because they are intermixed with some moderate opposition elements America supports. Now the U.S. would benefit from Moscow's inroads into targeting the group.

While the draft document primarily focuses on al Qaeda's Syrian affiliate, it does say that the cooperation may also facilitate strikes against ISIS, also known as Daesh, but it adds that both countries "reserve the right to conduct unilateral strikes against Daesh targets outside of designated areas."

The vast majority of U.S. airstrikes in Syria have been targeted against ISIS while U.S. officials have repeatedly accused Russia and its close ally Syria of largely striking opposition forces, some of which are backed by the U.S. Kerry said last month that reaching an understanding with Russia was "the most important thing" in ending the bloodshed in Syria.

But the Pentagon wants some assurances the Russians will live up to the terms of any agreement in which the U.S. makes concessions. Defense officials told CNN that, internally, Carter has expressed skepticism the Russians can be trusted.

The Russians would gain a critical win they have been seeking for months: intelligence about where U.S. aircraft are operating.

And U.S. officials said that a cooperative agreement with Russia could lessen the pressure the U.S. has been putting on the Russians to get Assad out of office. That is another key win for Moscow.

The U.S. still wants to see Assad encouraged to leave, although U.S. officials can't really say how that is more than a hope.

But from the Pentagon point of view, an agreement with the Russian military may be viewed as a loss. Defense officials have made it privately known that Carter is very skeptical the Russians would live up to their promises. And officials say the Pentagon still would not want to share the most sensitive information with Moscow.

Last month Carter publicly spoke about his concerns, using another name for ISIS.

"The Russians got off on the wrong foot in Syria. They said they were coming in to fight ISIL," he said. "And that they would assist the political transition in Syria towards a post-Assad government that could run the country and put that terribly broken country back together and give the people the future they deserve."

He concluded, "They haven't done either of those things."

Carter said an existing agreement with the Russians on air safety procedures over Syria will continue. But beyond that, he remains publicly cautious when asked about additional sharing of airstrike information.

"If the Russians would do the right thing in Syria, and that's an important condition, as in all cases with Russia, we're willing to work with them," he said. "That's what we've been urging them to do since they came in."

Still, there is a growing sense in Washington that al-Nusra must be dealt with in Syria. One reason: If the U.S.-backed rebels can push ISIS out of their Syrian stronghold in Raqqa, there is concern al-Nusra could rapidly move into some vacuum there.

While the State Department has not publicly discussed the idea of joint military operations, responding to a question from CNN department spokesman John Kirby told reporters Monday that "we continue to explore options and alternatives and proposals with respect to the fight against Nusra and Daesh in Syria."

"The degree to which the Russian military is willing to be committed to the fight against those two groups and exclusively those two groups, well, that's a conversation that we're willing to have," he added.

When asked about the possible agreement by CNN's Jill Dougherty, Kremlin spokesman Dmitry Peskov said Thursday, "First we have to learn [about the proposal] from [Kerry], not from the paper."