Timur Siraziev was seen repeatedly passing the 77th Brigade’s barracks in Hermitage, Berkshire, the Mail on Sunday reported.
The newspaper said he had been recorded by the base’s security cameras and had been seen filming close to the perimeter fence.
Mr Siraziev is listed on the Russian Embassy website as bureau chief for television station Channel One.
A senior Army source said: “Timur Siraziev’s suspicious behaviour was monitored by the base’s security systems throughout.”
Defence Secretary Gavin Williamson said: “We take the security of our bases and personnel incredibly seriously.
“If a member of the public sees anyone acting suspiciously in or around a military base it should be reported to the police as a matter of urgency.”
The news comes amid heightened political tensions between the UK and Russia after Moscow’s GRU military intelligence agency was blamed for the poisoning of former spy Sergei Skripal and his daughter Yulia in Salisbury.
Last month, the new head of the British Army, General Mark Carleton-Smith, warned Russia posed a far greater threat to national security than Islamic terrorist groups such as al-Qaeda and Isil.
Modelled on the Chindits, a specialist unit which fought in Burma during the Second World War, the 77th Brigade was set up in 2015 comprising soldiers with propaganda, political activism and social media skills.
At the time, the Army said the regular and reserve unit was being created “to draw together a host of existing and developing capabilities essential to meet the challenges of modern conflict and warfare”, adding: “It recognises that the actions of others in a modern battlefield can be affected in ways that are not necessarily violent.”