UK non-profit organisation Presidium Network said Paul Urey (pictured) and Dylan Healy were captured on Monday at a checkpoint south of the city of Zaporizhzhia in south-eastern Ukraine.
Mr Urey, 45, from Manchester, was described by Mr Byrne as a “family man with children. His mother, Linda, said the family “were extremely worried” about Mr Urey, who is diabetic and needs insulin.
Mr Healy, a chef from Huntington, Cambridgeshire was driving the car when the pair were captured by the Russian military at a checkpoint, the Presidium Network said.
The Foreign Office said it is urgently seeking more information.

* A journalist for Radio Liberty in Ukraine has been killed during Russian shelling of Kyiv, the station has said.
Vera Gyrych (pictured) was at home when a missile hit her building, according to a statement from Radio Liberty.
She was described as “a bright and kind person, a true professional” by her employer.
A wonderful person is gone,” her colleague Oleksandr Demchenko said on Facebook.
Radio Liberty, also known as Radio Free Europe, is a US-funded organisation which broadcasts news in areas of the world where free press may be restricted or not yet established.
Earlier, Kyiv mayor Vitali Klitschko said one body had been recovered after yesterday’s missile attack, though no further details of the person killed were given at the time.
Moscow has confirmed that it hit Ukrainian targets, but has not commented on the strike on the building.
One person has died following yesterday’s Russian missile strike in Kyiv, the city’s mayor has said.
In a post on his Telegram channel, Vitali Klitschko said rescuers had found a body while searching through the rubble of a residential building that was hit by a rocket in the Shevchenkivskyi district.
The strikes took place while United Nations Security General Antonio Guterres was in the capital to meet Ukraine’s President Volodymyr Zelensky.
Zelensky accused Russia of trying to humiliate the UN.