One of the men is charity medic Kevin Cornwell, 53, from Middlesbrough, according to the Presidium Network, a UK-based non-profit organisation that supports communities in crisis.
It said Cornwall and another unnamed man were arrested by the Taliban in January.
The third man, who was arrested on a different date, is understood to be Miles Routledge, a self-styled "danger tourist".
Who is Miles Routledge?
Routledge, the self-styled "Lord Miles", is a 23-year-old from Birmingham who spends his time travelling to the world's most dangerous places.
In his Twitter bio, the former Loughborough University physics student states: "I go to the most dangerous places on Earth for fun!"
He has more than 150,000 followers on Twitter and more than 59,000 subscribers on his YouTube channel.
Where has Miles Routledge visited?
Routledge travelled to Ukraine shortly after it was invaded by Russia last year, and has also visited South Sudan, Chernobyl and Brazil's Snake Island.
Videos posted on his YouTube channel also show him crossing the US/Mexico border and spending two days homeless in New York City.
Hasn't Routledge been to Afghanistan before?
Yes. Routledge was evacuated from Afghanistan by British Armed Forces in August 2021 after the Taliban seized control of the country.
He had posted updates on social media about his travels after the Taliban takeover.
Watch: Government 'in negotiations' to free Britons held by Taliban, says home secretary
Routledge said he chose Afghanistan because he enjoyed "dark" and "extreme" tourism.
Routledge was hailed by his supporters but criticised in some quarters for his trip.
He claimed he encountered Taliban members during his trip and that he had seen dead bodies in the street and shielded from gunfire in a Nato safe house in Kabul.
He was extracted to Dubai in what he described as a British Army plane.
He posted at the time: “The happy ending: landed in Dubai thanks to the brilliant people at the British army. All safe!”
He posted updates about the trip on Facebook, streaming site Twitch and the 4chan discussion board.
He travelled to Kabul as an alternative to a beach holiday, having looked up the most dangerous places to visit in the world and picked it off a list.
Has there been any contact with the three men?
Routlege's mother, Susan, said she has not been allowed to speak to him.
She told the Mail on Sunday: "I'm distraught. It is so worrying. I just want help to find out what has happened to him and make sure he is okay.
"If I can get a message to those who are holding him, I just want to ask them to tell him he has my love and support.
"I am just desperate to know that he is safe. He keeps travelling because he loves it, he's only young and he's just finding himself after finishing university. I just want him to keep safe."
The families of Cornwell and the other unnamed man have been able to speak to them in an "emotional" call.
Scott Richards, co-founder of the Presidium Network, said on Twitter: “We can confirm the men have spoken with the family, the conversation was unscripted, and that they are being treated fairly.
“The family was able to speak for one minute to one minute and a half and to speak freely, clearly what is an important and emotional call.”
Richards said the call “represents tremendous progress in the situation”, adding: “The details of that call are private but we understand a great relief to the family.
“The relief Kevin’s family expressed after hearing his voice for the first time in three months not knowing if he was well brought a sense of peace and gave them hope that this situation will be resolved soon.”
What is being done to help the three Britons?
On Sunday, home secretary Suella Braverman suggested the government is “in negotiations” regarding the three men.
The two men being assisted by Presidium were detained by Taliban secret police on 11 January.
Presidium said the two men are being held because of a "misunderstanding" over a missing licence for a weapon in a safe in Cornwell's room.
It is unclear how long Routledge has been held for.
Braverman told Sky News: “Anyone travelling to dangerous parts of the world should take the utmost caution.
“If there are risks to people’s safety, if they’re a British citizen abroad, then the UK government is going to do whatever it takes to ensure that they’re safe.
“And the government is in negotiations and working hard to ensure people’s safety is upheld.”
Pressed on whether negotiations are currently happening, Braverman replied: “If there are problems and if there are safety concerns to British individuals abroad then the FCDO (Foreign, Commonwealth and Development Office) will be working actively to ensure people are safe.”
Richards said the men are being held in a general directorate of intelligence facility for foreigners.
An FCDO spokesman said: “We are working hard to secure consular contact with British nationals detained in Afghanistan and we are supporting families.”