Non-profit group, the Presidium Network, said Paul Urey and Dylan Healey were detained early on Monday morning at a checkpoint south of the city of Zaporizhzhia in south-eastern Ukraine.
Mr Urey and Mr Healey travelled to Ukraine of their own accord as humanitarian aid volunteers, the organisation said.
Mr Urey's mother, Linda, who lives near Preston, is pleading for his safe return home, and added he has Type 1 diabetes and needs insulin.
She said: "My family and I are extremely worried.
"We know my son Paul and his friend who was a humanitarian aid volunteer in Ukraine have been captured by the Russians.
"He was out there on his own accord. We want everyone's support to bring my son home and pray he is safe.
"My son Paul is also Type 1 diabetic and needs his insulin.
"We have asked the Presidium Network to help us and also the FCDO to help as well. We pray for him and hope he is safe."
Mr Urey, who is 45 and from Warrington, and Mr Healey, 22, from Cambridgeshire, were not working for the Presidium Network, which helps to get aid into Kyiv.
The organisation said the pair were driving to help a woman and two children to evacuate when they went missing.
Presidium Network said it is concerned Russian forces may think the two men are British spies.
Dominik Byrne, one the founders of the Presidium Network, said: "Basically what needs to be done is two things - we need to put pressure on the Government to take this case seriously, and try, through their networks to verify this, but also to kind of help us find these people.
"But also because I know diplomatic channels are completely broken down, we use these tactics to find people by having it very public and putting pressure publicly on Russia to determine that they have got these two people and that they're safe and well.
"And we want to tell the Russians basically that these aren't spies. These aren't military people. These are just humanitarian workers who got caught in a bad situation."
He said the men were mostly helping with the distribution of medical aid and also helping with evacuations.
Ms Urey told Sky news that her son going missing has been her "worst nightmare".
She said her son went back out to Ukraine about a week and a half ago after he'd come home for leave, adding: "I begged him not to because of ... Russia's bad."
She later added: "I said 'please don't go'. He said 'Muma I can't live with myself knowing people... need help to get to a safe place, I have to go. I would feel bad.'"
On when she last spoke to him, she said: "I was on FaceTime with him up to 4am on Monday morning and that was it - gone."
Ms Urey said her son messages and Facetimes her several times a day but now she is hearing "nothing".
"Something's wrong - they've got him, definitely. He would contact me if he could and he can't," she said.