American forces engaged Pro-Assad troops with artillery, rockets and air strikes on Tuesday following what CENTCOM has characterized as an "unprovoked attack" against USA backed forces in Syria.
Syrian President Bashar al-Assad, who is supported by Russian Federation and by Shi'ite militias backed by Iran, has said he wants to take back every inch of Syria.
The coalition supports the Kurds in driving out Islamic State militants from Syria.
Syria's Deir Ezzor hosts a patchwork of rival forces and their worldwide partners, from Russian-backed government troops and US-backed forces to Shiite militias and jihadists.
"We suspect Syrian pro-regime forces were attempting to seize terrain SDF had liberated from Daesh (Islamic State) in September 2017", the USA official said.
One US official told Reuters, "We suspect Syrian pro-regime forces were attempting to seize terrain SDF had liberated from [IS] in September 2017", adding that they were likely seeking to seize the oilfields in Khusham, a major source of revenue for IS from 2014 till it recaptured by the SDF.
In a letter to the United Nations, Syria's foreign ministry described the strike as a "war crime" and called for the coalition to be dismantled, Syrian state news agency SANA said.
Bashar al Assad often refers to any group that stands in opposition to his formal government as terrorists, as the nation has been embroiled not only in a fight against the Islamic State, but an ongoing civil war. Pentagon officials, however, do not believe the Russians were after the Americans.
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The Turkish officials said two leaders also agreed that efforts to create "observation posts" in Syria's Idlib province as part of a "de-escalation" agreement between Turkey, Russia and Iran should gain momentum.
The flare-up lasted into the small hours of Thursday and was sparked by an attack on positions held by the Kurdish-led Syrian Democratic Forces (SDF) in Deir Ezzor province. There was no immediate reaction from Washington over Moscow's allegation.
The air raids coincided with escalating violence elsewhere in Syria. Mattis said the USA was still trying Thursday to establish exactly who the attackers were but added that the Russians have said they did not have forces there and "you can't expect somebody to deconflict something they can't control". A USA official said that "20 to 30 artillery and tank rounds landed within 500 meters" of the SDF headquarters.
United States officials say that the continued presence of coalition advisers in support of the SDF is essential to prevent the re-emergence of ISIS, and that there may be two or three months more fighting against ISIS remnants in remote areas close to the border with Iraq.
The U.S. -led coalition was set up in 2014 to battle Islamic State fighters in both Syria and Iraq, who were largely defeated a year ago.
Neither U.S. officials nor the U.S. -backed coalition have offered details on the identity of attacking forces.
Russia's foreign ministry meanwhile claimed Thursday that the USA airstrike on pro-Assad troops reflected Washington's attempts to seize Syrian economic assets.
Russia's Defence Ministry said the pro-government militias involved in the incident had been carrying out reconnaissance and their activities had not been previously agreed with Russian Federation.