Iraq-born Bisher al-Rawi, who had lived in Britain since 1985, and Jamil el-Banna, his Jordanian business partner granted refugee status in Britain in 2000, are alleged to have been associated with al-Qaida through their connection with the London-based preacher Abu Qatada.
Al-Rawi and el-Banna were arrested several years ago in Gambia while trying to return to Britain with electronic equipment the authorities described as suspicious.
The men's lawyers say the equipment was a battery charger. The lawyers have said the two men were arrested after British intelligence passed on information about their travel plans to US intelligence.
The Newsnight programme on BBC TV on Monday night said it had obtained secret telegrams that British intelligence had sent to US officials leading to the arrests of al-Rawi and el-Banna in Gambia.
The BBC said the telegrams do not indicate whether the British government knew that the CIA's arrest and interrogations in Gambia would lead to the men being flown to Guantanamo and held there.
British officials sent the first telegram to the CIA on 1 November 2002 when the two men were arrested at Gatwick Airport, saying one was carrying an object that could be used as part of a car bomb, the BBC said.
The men were taken from Gambia
to Egypt, to Afghanistan and Cuba
The men were released when British intelligence found the device to be harmless, but the Americans were not told this, said Newsnight.
The following week, al-Rawi and el-Banna flew to Gambia, and British intelligence sent a telegram to US officials, giving them the flight details and reminding them of the men's association with Abu Qatada, the BBC said.
Three days later, another telegram told the Americans that al-Rawi was an Iraqi Islamic extremist who was a member of Abu Qatada's close circle, whose behaviour and finances had been suspicious and who had experience as a diver and a parachutist, Newsnight said.
After being arrested at the airport in Gambia, the men ended up in the custody of the Americans. From there, the CIA took them on a "rendition" flight to Cairo, Egypt, where the plane refuelled, then to a CIA facility in Afghanistan, where they were held and interrogated as suspected terrorists, the programme said.
In early 2003, the men were transferred from Bagram Air Base north of Kabul to Guantanamo, Cuba, where they remain prisoners.
Newsnight said the telegrams prove Britain's involvement in the men's arrests.
It was not immediately possible to get the British government's reaction to the BBC report.