In a statement to CBS News, a foreign ministry spokesman added: "Aamer's case remains a high priority for the UK Government, and we continue to make clear to the U.S. that we want him released and returned to the UK as a matter of urgency."
Several British members of Parliament wrote to members of Congress two years ago calling on the U.S. to free Aamer. They never received a response, according to British Member of Parliament Jeremy Corbyn, who co-authored the letter.
"He has a legal right to be in the U.K. His wife is a British national, his children are British nationals, and therefore, we believe, that the British government has a responsibility to care for him," Corbyn said. "Why don't they either put him on trial or release him? I suspect they think he knows too much, will say too much, and will describe too much about life in Guantanamo."
Aamer does claim he endured physical abuse at the hands of his American captors at Guantanamo and earlier at Bagram Air Base in Afghanistan - that he was forced to stand naked before groups of guards, including women, was deprived of sleep, and exposed to loud music, bright light, and temperature extremes. He says at times he was beaten, slammed against walls, and suspended by his arms with his arms twisted behind his back.
At Guantanamo, Aamer has protested his confinement by going on hunger strikes, resulting in guards force-feeding him with tubes down his nose.
Aamer has told his attorneys that he is among the growing group of active hunger strikers at Guantanamo. On Friday, Aamer told Clive Stafford Smith, Director of the Human Rights Group Reprieve, that he has been refusing meals since February 15 and has lost 32 pounds. The U.S. military does not release the names of any of the 37 men it counted as hunger strikers as of Friday.
Aamer spends 22 hours a day alone in a steel cell in Guantanamo's Camp Five Echo and gets two hours a day for outdoor recreation by himself, according to Kassem, who most recently visited Aamer last month.
"He is alone at all times. He is allowed no social contact with other inmates. The U.S. military will tell you that is not isolation, because he can scream through the vents and perhaps another prisoner will hear him and respond," Kassem said.
Guantanamo detainees are not permitted visitors except attorneys who have passed extensive background investigations. Phone calls to families do happen occasionally. Outgoing and incoming mail is subject to military censors.
"I miss you a lot," Aamer's 12-year-old son, Saif, wrote to his father last year in one redacted letter seen by CBS News. "We all love you a lot too."
The boy listed things he likes as rugby, basketball, swimming, wrestling, skating, karate, computer games, science, reading, and the Koran.
"Please come back quick so we can go to places with you and so we can see you," Saif wrote. "We all have grown."
With no criminal or military charges ever filed against him, the only available government accusations about Shaker Aamer are contained in old Defense Department detainee assessments posted by WikiLeaks. The 2004 paperwork alleged Aamer was tied to a European support network for al Qaeda and lived briefly in London with convicted 9/11 conspirator Zacarias Moussaoui. It called Aamer a close associate of Osama bin Laden who had visited an al Qaeda training camp in Afghanistan. It said he later participated in jihadist combat and had expressed a desire to be a martyr.
"What you have in those documents is not evidence," said Kassem.
Jeremy Corbyn, the British MP advocating for Aamer's return, called on President Obama to fulfill his 2008 campaign pledge. "He told the world he would close down Guantanamo Bay, because he thought it was a disfigurement of the principles of U.S. democracy and justice. I want him to stand by his words."
Before he left his post this month, National Security Council spokesman Tommy Vietor told CBS News: "The President is still committed to closing the prison at Guantanamo Bay. He thinks it is in our national security interest to do so. But the bottom line is Congress has put every hurdle it can find in our path to close the prison."
Congress has also blocked detainee transfers to the U.S. mainland for either incarceration in federal prisons or trials in federal courts, Vietor said, "despite the fact that federal courts are a more tried and tested venue for terrorist conspiracy and material support charges."
Still, Ambassador Dan Fried, the U.S. State Department's special envoy for transferring Guantánamo detainees during Obama's first term, has been re-assigned, and his office was eliminated with its duties now handled by the department's legal counsel's office.
The word "Guantanamo" never came up during Senate confirmation hearing for newly sworn-in Defense Secretary Chuck Hagel.
Aamer's supporters launched an online petition for his release on the U.S. government's "We The People" website but fell short of the 100,000 signatures that were needed by March 8 to require the White House to formally respond.
"He's a very perseverant person, he's a very hopeful person," Aamer attorney Kassem said. "He is by no means a broken man. He continues to struggle and hope and live."CBS News correspondent Mark Phillips and producer Ella Flaye in London contributed to this report.