The reports attribute the alleged casualties to a recent United States airstrike and put the figures between 100 and over 600.
The clash is believed to be the deadliest incident between United States and Russian citizens since the end of the Cold War. These claims can not be verified.
Earlier, the USA military said that it killed more than 100 Syrian militiamen allied with the government of President Bashar Assad after some 500 fighters targeted "well-established Syrian Democratic Forces headquarters" in Deir al-Zor province in what the Pentagon called an "unprovoked attack".
The reports cited activists who said at least four Russian citizens were killed in Syria on Wednesday.
Syrian state media have reported that the strikes left dozens killed, accusing the USA of a "brutal massacre".
They were named as Alexi Ladigin, of Ryazan, Stanislav Matviev, Igor Kostorov and Vladimir Loginov, of Kaliningrad.
"The tank had been maneuvering with coordinated indirect fire on a defensive position occupied by Syrian Democratic Forces and Coalition advisers".
He also added that the Euphrates river has always been used as a line of demarcation between USA forces and the opposition, and the troops of Russian Federation and the government of Syria.
Inflation rate remains steady at 3%
It's a sharp shift from past year when consumer price growth slowed, even as the economy grew and the unemployment rate fell. Markets would greet an increase in consumer inflation that is in line with expectations with "relief", Jones says.
The U.S. -backed Kurdish-led forces have been vying with Russian-backed Syrian troops reinforced by Iranian-supported militias for control of the oil-rich Deir el-Zour province.
The Minister of defence of the USA James Mattis said that did not have precise information that the Russian mercenaries have been eliminated during an air strike by forces of Syrian dictator Bashar al-Assad. One U.S. official and three Russian officials familiar with the incident all corroborated it with Bloomberg, telling the outlet of the Syrian forces' failed attack against a U.S. -held military base and refinery.
Russian Federation has waged a military campaign in Syria since September 2015, helping Syrian President Bashar Assad's government seize large chunks of territory and turn the tide of war in its favor.
Putin's spokesman Dmitry Peskov said the reports needed to be verified. "As it came closer they were notified when the firing began", Mattis said. Ankara has repeatedly warned against the repercussions of using one terrorist group to defeat another, while the USA has been touting the "effective results" of its cooperation with the YPG in the fight against Daesh. Maxim Buga, a leader of the Cossack community there, said Loginov had been killed around February 7 along with "dozens" of other Russian fighters.
The incident took place five miles across the Euphrates River near Deir ez Zoir, which serves as a "de-confliction" line between U.S. and Russian/Syrian backed forces.
"Those steps look increasingly as part of efforts to create a quasi-state on a large part of Syrian territory from the eastern bank of the Euphrates all the way to the border with Iraq", Lavrov said, in remarks that highlighted the geopolitical jostling in Syria in the wake of the fight against Isis.
The US Defense Secretary Jim Mattis played down the incident, saying: "This could just be a local couple of guys doing something".