The two men, both in their 60s, had just been dropped off by the helicopter when it went to take off and tipped onto its side, killing them instantly.
Their Russian guide also died, but the pilot of the French-built Eurocopter EC-120 survived.
The incident happened in the far northern Kola peninsula at Rynda River, around 90 miles east of the Arctic city of Murmansk, at 8am UK time.
The area is said to attract wealthy tourists who are drawn to its salmon-rich waters.
One Russian news website report said the pair, who have not yet been named, were ‘VIP tourists’ who were on a special fishing trip to some of the world’s remotest rivers.
“We are aware of reports of the deaths of two British nationals in Russia,” a Foreign Office spokesman said.
“We are in touch with the local authorities and stand ready to provide consular assistance.”
A spokesman for manufacturers Eurocopter also said it was ready to provide any support needed.
“We are awaiting more information from Russia and are ready to give any support that is needed,” he said.
One Russian news agency quoted a security source who said the men were staying in a “luxury camp for fishing”.
They reported that the men were most likely killed by the helicopter’s blade when it tipped onto its side.
In a separate incident, British and French emergency teams were called to the English Channel after a small aircraft crashed into the sea four miles from the UK coast.
The American-registered plane with one person on board came down about 15 miles off Dungeness, in Kent, yesterday at around 2:30pm.
A cross-Channel search and rescue operation was launched involving both English and French teams, including a French helicopter and the Dungeness RNLI lifeboat. The alarm was raised after the single-engine light aircraft, which was heading from London to Le Touquet, was overdue, a spokesman for the Maritime and Coastguard Agency (MCA) said.
The incident happened four miles inside UK waters and is being co-ordinated by the British with help from the French authorities.
An MCA spokesman said last night: “We had reports that an aircraft was overdue. It was heading from London to Le Touquet.
“The Dungeness lifeboat is en route and we believe the Calais lifeboat is also involved.
“The search is ongoing at the moment to find the pilot.”