The mother of an American military veteran who volunteered to defend Ukraine from Russian invaders and was reportedly captured recently said a video of him expressing his love for her gave her “great hope”.

Lois Drueke told the Guardian on Friday that she believes the video of her son Alexander Drueke – distributed by Russian state media – is authentic because of a phrase he uttered in the clip with his distinctively deep voice.

Lois Drueke declined to specify what that phrase was, but in the video, her 39-year-old son is seen saying: “Mom, I just want to let you know that I’m alive, and I hope to be back home as soon as I can. So love diesel for me. Love you.”

She added that the US state department, which is investigating the capture of Drueke and at least one other American volunteer in Ukraine, had still not officially confirmed the video’s authenticity. But, she acknowledged, “We think it’s Alex – it looks like him, [and] it sounds like him.

“Just seeing him move, just hearing his voice, gives us great hope. I don’t want to get my hopes up too much, but it does seem to be him.”

The brief video clip in question began circulating after US president Joe Biden told reporters that he had been briefed on the disappearances in Ukraine of Drueke, his fellow military veteran and Alabama resident Andy Tai Ngoc Huynh, and Grady Kurpasi, who apparently went missing in April.

“We don’t know where they are,” Biden said to reporters. “But I want to reiterate: Americans should not be going to Ukraine now.”

Biden’s remarks came a day after a photograph surfaced that appeared to show both Drueke and Huynh in the back of a military truck with their hands tied behind their back, the clearest piece of evidence that they were no longer missing but had been arrested by Russian forces invading Ukraine.

That photo was followed on Friday with the video apparently depicting Drueke addressing his mother, along with another clip in which he said in English, “I am against war.”

Videos of Huynh also surfaced, including one with Russian state media branding in which he calmly recounts a purported version of events on the day he and Drueke were captured.

Huynh, 27, said he and Drueke were part of a unit with a large number of French nationals that became locked in a gunfight with Russian troops.

“The Ukrainians were retreating, and we were asked to cover their retreat,” Huynh says in the clip. “When we were covering them, the Russian forces overran our position.”

Huynh said he and a colleague waited for three hours “in a fighting hole just to make sure the coast is clear”. After emerging, the pair walked five more hours through the woods and on to a road where they “surrendered” to Russian forces, Huynh said on the video.

The Telegraph, which was first to report that Drueke and Huynh had gone missing, cited an account from an unnamed fellow fighter who said the pair were taken prisoner after a battle with Russian forces north-east of Kharkiv on 9 June.

Lois Drueke cautioned against taking everything in the videos at face value. She said her son – before going to Ukraine – had warned her that he could be taken prisoner and, if that happened, might be made to say inauthentic things on video by his captors.

Many have called on people to not share videos of those taken captive during the Ukraine-Russia war, arguing that they violate the Geneva conventions’ prohibition against making a spectacle of imprisoned combatants. But media organizations can report on such images without breaching the conventions, which apply only to nation states or other so-called detaining powers.

Drueke, Huynh and Kurpasi are believed to be among hundreds of Americans – and possibly more – who went to Ukraine in hopes of helping the country repel the Russian invasion that began in February.

Drueke served two combat tours in Iraq and was teaching Ukraine’s troops how to use American-made weapons. Huynh and Kurpasi each previously spent time with the US Marines.

Captured American-born defenders of Ukraine create a diplomatically tricky situation for the US. The US is pumping billions of dollars into supporting Ukraine’s defense but is avoiding a direct clash with Russia.

Drueke’s aunt, Dianna Shaw, on Friday pleaded with the captors of her nephew and Huynh to treat them “humanely”.

She also said that a bipartisan congressional delegation from Alabama was working hard to keep his and Huynh’s loved ones informed about their apparent capture. Shaw singled out the aides of both Democratic House representative Terri Sewell, of Huntsville, and Republican senator Richard Shelby, of Birmingham, as being particularly open about where things stood.

“We are very encouraged by the way all … have communicated consistently with us and with each other,” Shaw said.