If confirmed, the news from Tuesday could further inflame tensions between Russia and the US.
Russian media also cited unconfirmed claims that overall casualties could be as high as 200, the majority Russians. These claims cannot be verified.
Russian media said the Russian private contractors were part of pro-government forces that advanced on oil fields in the eastern Deir el-Zour province and were targeted by the United States.
The reports cited activists who said at least four Russian citizens were killed in Syria on Wednesday.
Citing Natalya Krylova, a municipal lawmaker in the town of Asbestos in the Urals, the state news agency Tass said local residents Igor Kosoturov and Stanislav Matveyev were killed in the strike.
Quoting the Cossack group in the westernmost Kalningrad region, the Interfax agency said a member named Vladimir Loginov was killed in combat in the province of Deir el-Zour.
It also quoted Alexander Averin, leader of extreme leftist group Another Russia, as saying that one of its activists, Kirill Ananyev, was killed by the U.S. strike near Khusham.
The U.S. strike came as U.S.-backed Kurdish-led forces clash with Russian-backed Syrian troops, reinforced by Iranian-supported militias for control of the oil-rich Deir el-Zour province.
Last week, the US military said it unleashed air and artillery barrage on Syrian government-backed troops after an estimated 300 attackers launched what appeared to be a coordinated assault on Syrian opposition forces accompanied by US advisers in the province of Deir el-Zour. It said it was in regular communication with Russian counterparts before, during and after the attack and that Russian officials assured coalition officials they would not strike coalition forces.
It said the US strikes, which reportedly killed around 100 of the attackers, were in self-defence.
Russia 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.
What Russia has said
President Vladimir Putin's spokesman, Dmitry Peskov, refrained from commenting on reports about the incident which happened last week. He says they need to be verified.
He said the situation in Syria was not discussed during a phone conversation between Putin and US President Donald Trump on Monday.
The Russian Defence Ministry has insisted its troops were not involved in the incident. It says 25 Syrian volunteers were wounded in the US strike.
It also noted pro-government forces that advanced toward the area under control of the US-supported forces had failed to coordinate their action with the Russian military in Syria.
The Russian Defence Ministry says the incident on February 7 reflects a US push to grab Syria's economic assets under the cover of fighting ISIL.
Without mentioning the US strike, Russian Foreign Minister Sergei Lavrov said "Americans have taken dangerous unilateral steps".
"Those steps look increasingly as part of efforst to create a quasi-state on a large part of Syrian territory, from the eastern bank of the Euphrates River all the way to the border with Iraq," he went on.
What the US has said
Defence Secretary Jim Mattis says Russia has told the US there were no Russians in the area of the strike. Mattis said he saw the news reports about Russian contractors possibly being casualties of the bombing.
"I don't have any reporting that some non-Russian Federation soldiers, but Russian contractors were among the casualties," he told reporters. "I can't give you anything on that. We have not received that word at Central Command or the Pentagon."
Last week, Mattis dismissed any suggestion that Russia has any control over the attacking force. He said he could not identify nationalties and motives.
How significant is this?
Very, potentially. During the war in Syria, Russia and the US have maintained a communications channel to avoid collisions between their warplanes and soldiers on the battlefield.
If Russian deaths from the US strike are confirmed, it would mean the two countries were on the verge of a direct military confrontation in Syria.
The episode could further strain ties between the two countries. They are already under pressure over the Ukrainian conflict, the war in Syria and the alleged Russian meddling in the 2016 US presidential election.
Private contractors - why are they in Syria?
Russian media have reported that, in addition to Russian forces deployed to Syria, thousands of Russians have also fought there as private contractors, most serving with the Wagner Group. The private fighters allowed the Kremlin to keep the official death toll from its campaign in Syria low and avoid the negative public perception of its involvement in Syria. Vladimir Putin is running for re-election on March 18.
The Kremlin and the Russian Defence Ministry have refused to comment on media reports on Russian private contractors in Syria.
The US has used private contractors in Iraq and Afghanistna for years.
However, Russian law forbids the hiring of mercenaries or working as one. Some Russian lawmakers have called for legislation that would set rules fro private military contractors.
What they are saying
"If many Russian citizens were killed, officials, including the commander-in-chief, must tell the nation about it and determine who is reponsible for that," - Liberal politician and presidential candidate Grigory Yavlinsky