Asked about reports of Mr Anderson’s capture, David Lammy said he had “been updated about that development in the last couple of days”.
Scott Anderson, Mr Anderson’s father, said he had begged his son not to go to Ukraine.
“He wanted to go out there because he thought he was doing what was right,” Mr Anderson told the Daily Mail.
“I’m hoping he’ll be used as a bargaining chip, but my son told me they torture their prisoners and I’m so frightened he’ll be tortured.”
Mr Anderson said he was sent the video by his son’s commander and was left “in complete shock and tears”.
“I could see straight away it was him. He looks frightened, scared and worried,” he added.
In the video, first posted to the Telegram messaging platform, Mr Anderson tells a man questioning him from behind a camera that he served as a private in the British Army from 2019 to 2023.
He says he joined the Ukraine’s International Legion – a military unit made up of foreign volunteers – after losing his job and seeing reports on television about the war.
He says he flew to Krakow in Poland from Luton and travelled from there by bus to the Ukrainian border.
Ukraine launched a surprise attack into Kursk on 6 August, advancing up to 18 miles (29km) and taking control of around 1,000 square kilometres of Russian territory.
Russia has deployed some 50,000 troops to the region, and has begun retaking territory amid fierce fighting.
The Foreign Office said it was “supporting the family of a British man following reports of his detention”.