The United States and Russia announced a new multi-step plan to bring Syria closer to a negotiated peace deal. (Jason Aldag,Karen DeYoung/The Washington Post)

Hours after reaching an agreement on Syria last Friday with Secretary of State John F. Kerry and clearing the final deal with Moscow, Russian Foreign Minister Sergei Lavrov wandered the halls of their meeting venue in Geneva, waiting for Kerry to get the okay from Washington.

In a secure room upstairs, a frustrated Kerry was on hold. Already deep into a conference call with President Obama’s top national security team, he was waiting for the Defense Department to locate its legal counsel to sign off on one of the many provisions of the accord that Defense Secretary Ashton B. Carter was questioning.

Obama, who did not attend the principals’ meeting, ultimately approved the agreement, and a news conference was held at midnight, Geneva time.

But beneath the politics and diplomacy of the deal — which began with a cease-fire Monday, to be followed, if it succeeds, by coordinated U.S.-Russian counterterrorism airstrikes — the prospect of military-to-military cooperation does not sit well with the Defense Department.

Secretary of State John F. Kerry and Russian Foreign Minister Sergei Lavrov confer in Geneva on Sept. 9 at the end of a joint news conference following their meeting to discuss the crisis in Syria. (Kevin Lamarque/AP)

“There is a trust deficit with the Russians; it is not clear to us what their objectives are,” Gen. Joseph L. Votel, head of the U.S. Central Command, said Wednesday. “They say one thing, and we don’t necessarily see them following up on this.”

That mistrust resides most deeply in Carter, who officials familiar with the Russia negotiations said almost single-handedly delayed Friday’s final agreement with his repeated questions during the conference call. A representative for Gen. Joseph F. Dunford Jr., chairman of the Joint Chiefs of Staff, voiced little objection during the principals’ meeting, officials said.

But Pentagon officials acknowledged widespread concern that Russia will not live up to its end of the deal, and they fear that the U.S. military will be blamed for problems or the failure of an initiative it does not fully support. Many are still smarting from criticism and derision over an earlier program to build up an army of U.S.-trained Syrian fighters that repeatedly stumbled and ultimately was abandoned.

Kerry and other proponents of the agreement say that they are similarly wary of Russian intentions but that they do not see another way to stop the civil war in Syria, while retaining focus on what the administration views as the far more important battle against the Islamic State and without a major U.S. intervention that Obama has repeatedly rejected.

“What’s the alternative?” Kerry asked in an interview with NPR this week. “The alternative is to allow us to go from 450,000 people who’ve been slaughtered to how many thousands more? That Aleppo gets completely overrun? That the Russians and [Syrian President Bashar al-] Assad simply bomb indiscriminately for days to come, and we sit there and do nothing? That’s the alternative to trying to get this done, if America is not going to go in with their troops — and America’s made the decision we’re not going in with our troops.”

Amid reports of internal administration clashes, and after a terse and somewhat grudging initial Pentagon statement saying that “we will be watching” the Russians to make sure they comply, the White House said Wednesday that Obama is not looking for “a bunch of people that have the exact same opinion.”

But the president, White House press secretary Josh Earnest said, is “entirely confident that once he’s made a decision, that he can count on the members of his team to execute that strategy with excellence.”

After commending Kerry for the deal, Carter said at a Wednesday news conference in Austin, “We in the Defense Department will play whatever role we have with our accustomed excellence.”

Although many Republicans, and some Democrats, have criticized Obama for years for his reluctance to become involved in Syria’s domestic combat — and GOP presidential nominee Donald Trump has accused him of not listening to his generals — it is the generals who have been most averse to a direct role in the civil war.

“Dempsey was throwing himself under a bus in front of every idea that came out,” said a senior administration official, referring to former Joint Chiefs of Staff chairman Martin E. Dempsey, whom Dunford replaced last fall. The Defense Department “got the Iraq part, but it was the Syria thing — even the counter-ISIL part — that they didn’t want. My theory is that everyone in DOD understands Iraq, and they don’t understand Syria.” ISIL is an acronym for the Islamic State.

But military leaders share a deep-seated skepticism about whether the United States can change the course of the Syrian conflict at this stage, or, more broadly, can bring about lasting change anywhere in the Middle East.

During the months-long negotiations with Moscow that led to last week’s U.S.-Russia agreement, the Pentagon argued against more intervention and a diplomatic deal.

“Any collusion with Russia” has just made the long-standing reluctance to intervene worse, said the official, who spoke on the condition of anonymity to discuss internal deliberations.

Under the cease-fire terms of the deal, Russia is responsible for ensuring that the Syrian air force stops attacks against opposition forces and civilians, and the United States has pledged to do the same with those opposition groups it backs. If violence is substantially reduced for seven consecutive days — a period that began Monday night — and humanitarian aid begins to reach besieged communities, U.S. and Russian military and intelligence officials will establish a Joint Implementation Center.

There, they will share intelligence about the location of the two groups they both agree are terrorists and ineligible for a cease-fire: the Islamic State and the former Jabhat al-Nusra, which has renamed itself Jabhat Fatah al-Sham (Front for the Conquest of Syria) and claims to have severed ties with al-Qaeda. U.S. and Russian officials will discuss and agree on legitimate targets and determine which country is best situated to launch airstrikes against them.

U.S. military officials are working internally on ways to develop potential Front targets in areas where U.S. aircraft have not been recently active.

Officials from the U.S. Central Command, the European Command, and Dunford’s office, along with the Pentagon’s intelligence shop, are among those working out the details of how the Center would operate, including personnel, logistics and how to obtain the surveillance information required to support the expanded air campaign.

Commanders are keen to ensure that operations with Russia would not divert planes or other assets from current operations targeting the Islamic State, and that Russia would follow rules in its use of target information.

They are concerned about providing some legal protection to the Pentagon in the case of, for example, Russian aircraft striking a civilian target. To mitigate that risk, one possibility under consideration is a system in which the United States would develop a target, Russia would approve it and the United States would take the strike, and vice versa.

It’s not clear whether Russia, whose airstrikes have hit hospitals and civilian areas, will shift to using precision munitions in the jointly agreed strikes.

Under a measure lawmakers inserted into the defense appropriations bill after Russia’s ­military incursion into Ukraine, military-to-military cooperation with Russia is prohibited unless the defense secretary waives its provisions, notifies Congress and explains why the waiver is in the United States’ interest.

Congressional aides said the House and Senate armed services committees thought that such notification should have taken place last fall, when Washington signed a “deconfliction” memorandum with Moscow to ensure that their separate air operations against the Islamic State in Syria would not run into each other. The committees chose not to challenge a Pentagon assertion that the agreement was a “safety” issue and did not fall under the waiver requirement.

Following the new cease-fire and coordination deal, aides said, lawmakers have already requested waiver information from the Pentagon but have not yet received an answer.

Pentagon lawyers have determined that establishment of the Joint Implementation Center would trigger the notification and waiver requirement, although the Defense Department’s legal counsel, during last Friday’s conference call with Kerry, indicated that it did not have to be immediate.

Pentagon lawyers are still exploring whether and when, during the planning stage, notification has to take place. Some military officials are asking, for example, whether they are permitted to talk to the Russian military as they make plans to open the center.

Late Wednesday, a Pentagon statement said that senior Defense Department civilians and military officers had held a video conference with their Russian counterparts about ongoing in-flight safety “to avoid accidents and misunderstandings in the air space over Syria.”

The meeting, the statement emphasized, “was not part of separate discussions taking place on the possible stand-up of the joint integration cell.”

Thomas Gibbons-Neff and William Wan contributed to this report.

Correction: An earlier version of this article misstated who took part in a White House meeting on U.S. negotiations with Russia. It was a representative for Gen. Joseph F. Dunford Jr., the chairman of the Joint Chiefs of Staff, not Dunford himself.