The two haggard-looking men identified as British nationals Shaun Pinner and Aiden Aslin asked to be exchanged for Viktor Medvedchuk, a wealthy Ukrainian businessman close to President Vladimir Putin, who was recently arrested in the pro-Western country.
The broadcast did not specify where or by whom -- Russian forces or Moscow's separatist allies in eastern Ukraine -- the two men were being held.
The recordings were presented as interviews with journalist Andrei Rudenko, of the Russian state broadcaster VGTRK.
In the clip, Rudenko shows the two men a video published last week by Oksana Marchenko, Medvedchuk's wife, who demanded her husband's exchange for the two Britons.
The detainees then asked in English to be exchanged for the businessman.
In a statement released through Britain's foreign ministry, Pinner's relatives said the pair were "being held by the Russian army".
It added both families were working with the ministry "to ensure their rights as prisoners of war are upheld according to the Geneva Convention".
Pinner's family noted he was a "proud member" of Ukraine's marines, after moving to the country four years ago and marrying a Ukrainian.
This following "many years" in the British army, when he served tours in Northern Ireland and with the United Nations in Bosnia.
"In 2018 Shaun decided to relocate to Ukraine to use his previous experience and training within the Ukraine military," the family statement said.
"We would like to make it clear he is not a volunteer nor a mercenary, but officially serving with the Ukrainian Army in accordance with Ukrainian legislation."
Aslin's mother had appealed for his release in an interview with the British Daily Telegraph newspaper published on Friday.
The woman, Ang Wood, said that she had recognised her 28-year-old son from a Russian broadcast because of his distinctive tattoo.
"Aiden is a serving member of the Ukrainian armed forces, and as such is a prisoner of war and must be treated with humanity," Wood was quoted as saying by the newspaper.
In videos circulating on social media and carrying the logo of Kremlin-backed broadcaster RT, the young man appeared to imply that Ukraine is prolonging the conflict.
It was not clear whether the men had been forced to speak in the videos.
Ukrainian President Volodymyr Zelensky had proposed to exchange Medvedchuk, 67, for Ukrainians currently being held in Russia.
Asked about a potential exchange last week, Kremlin spokesman Dmitry Peskov stressed that Medvedchuk was "not a Russian citizen" and said he did not know if he wanted Moscow to interfere in his case.
On Monday, Peskov promised to "communicate" any response to the idea from Putin.