At least six UK residents are among about 210 detainees at the US military base who are refusing food, says Amnesty International and Reprieve.
They claim some detainees have been restrained and force fed.
They want ministers to seek a US pledge to allow independent observers access to the detainees, who are being held without trial or being charged.
The call came as US senators voted overwhelmingly to outlaw cruel or degrading treatment of detainees held in US custody abroad.
The US disputes the hunger strike numbers given by the human rights groups. It said 28 detainees are on hunger strike, from a high of 131.
“The camp is still closed to Amnesty International and the secrecy surrounding Guantanamo makes the hunger strike a frightening phenomenon,” said Amnesty’s director Kate Allen.
“We need to see the UK Government intervening to prevent deaths and injuries and to see that all detainees are either properly tried or immediately released in accordance with international human rights law.”
The detainees include Brighton law student Omar Deghayes, 36.
He was granted refugee status in the UK after fleeing Libya with his family in the 1980s when his father was assassinated.
Mr Deghayes has been held at Guantanamo Bay since being arrested while visiting Pakistan in 2002.
“Under the UN convention on refugees, the UK should be his surrogate state, protecting him as if he were one of their own citizens,” his sister Amani told the BBC News website.
“It’s quite outrageous how the government treat people they are suppose to be protecting.”
She said her family in the UK are feeling “really desperate” and fear for his life as the hunger strikes approach a third month.
Ms Deghayes is adamant her brother has no connection with terrorism.
“It’s been three-and-a-half years and nothing seems to have changed,” she added. “We still haven’t been told what evidence there is against him.”
The human rights groups have written to Tony Blair seeking assurances the government will assess the condition of the British residents on hunger strike.
Reprieve’s legal director Clive Stafford Smith is representing 40 Guantanamo Bay detainees.
“Conditions there at the best of times are disturbing,” he said.
“But to imagine my clients being held in four point restraints with a tube forced down their noses, after all that they have been through, just makes me sick.
“All these prisoners are asking for is that the US military abide by the Geneva Conventions.”
He said he had yet to receive a reply to a letter he wrote to UK Foreign Secretary Jack Straw four weeks ago on behalf of the British detainees.
“With each day that passes my clients’ health is deteriorating,” Mr Stafford Smith said. “We cannot wait any longer.”
Nine British nationals detained at Guantanamo Bay have been released without charge.
Of the other British residents at the camp, most are believed to have refugee status in the UK.
The Home Office said UK representation for people who have been granted refugee status “depends on the case in question”.
As for those non British nationals, who were resident in the UK before their detention, the Foreign Office said the government was “not in a position to provide consular or diplomatic assistance”. BBC