Russian President Vladimir Putin (R) and U.S. President Barack Obama share a toast during the luncheon at the United Nations General Assembly in New York, September 28, 2015. [Photo/Agencies]

UNITED NATIONS -- US President Barack Obama held "productive" talks with his Russian counterpart Vladimir Putin on Monday at the UN headquarters to discuss situations in Syria and Ukraine.

Speaking to reporters after the meeting, which lasted about 90 minutes, a senior US official said that "this was not a situation where either one of them was seeking to score points."

Before the meeting, the two leaders traded criticism over divergent positions on the Syria and Ukraine crises when addressing the UN General Assembly (UNGA) earlier in the day.

During their talks, Obama reiterated US support for Ukraine's sovereignty and territorial integrity and said a positive opportunity exists for the implementation of the Minsk accord in the next few months, the official said.

On the Syrian situation, the two sides fundamentally disagreed on the role that Syrian President Bashar al-Assad will play in resolving the civil conflict there, the official said.

He added that the Russians see Assad as a bulwark against extremists, while the Americans see him as continuing to fan the flames of a sectarian conflict there.

The meeting came amid US concerns about Russia's recent military buildup in Syria and after the announcement on Sunday of an understanding among Iran, Iraq and Syria to share intelligence about the extremist group Islamic State (IS), also known as ISIL.

The official said that the United States does not view Russia's military buildup in Syria as necessarily destructive to a positive outcome in Syria but rather the Obama administration's view on the Russians will depend on their actions going forward.

If the Russians use their military solely to fight the IS, that might be OK; but if they use their might to continue to strengthen Assad's battle against his own people, it will be negative, the official said.

The official also said the two presidents agreed that their militaries should communicate in order to "deconflict," or avoid military conflicts between them in the region.

Addressing the UNGA a couple of hours after Obama's speech, Putin said that supporting Assad was the only way to combat the IS.

"We think it's an enormous mistake to refuse to cooperate with the Syrian government and its armed forces who are valiantly fighting terrorism face to face," Putin said.

In an interview with CBS on Sunday, he also reiterated his support for al-Assad, saying that there is no other solution to the Syrian crisis than strengthening the effective government structures and rendering them help in fighting terrorism.

"We are considering intensifying our work with both President Assad and with our partners in other countries," Putin said.

Obama said in his UN address that the United States is prepared to work with any nation, including Russia and Iran, to resolve the conflict in Syria.

Meanwhile, calling Assad a "tyrant" and accusing him of killing tens of thousands of his own people, Obama said that the world must recognize that "there cannot be a return to the prewar status quo" in Syria.