UK lawyers acting for eight of the men said their clients had been held for up to 14 months without charge.
They compared it to the public's realisation of Guantanamo Bay and want the UK's High Court to free the men.
The MoD said detentions in Afghanistan were legal under the UN mandate.
Camp Bastion in Helmand province is the largest British military base in Afghanistan, housing nearly 30,000 servicemen and women.
Legal documents seen by the BBC suggest that an estimated 85 suspected insurgents are being held at the base in a temporary holding facility.
British forces in Afghanistan, operating as part of the International Security Assistance Force (Isaf), are allowed to detain suspects for 96 hours.
However, in "exceptional circumstances" - for example, to gather critical intelligence to protect lives - they can hold them for longer periods.
The Ministry of Defence has previously said that Isaf neither has the power nor the facilities to intern detainees in Afghanistan.
But nor can the British hand them over to the Afghan authorities, after Defence Secretary Philip Hammond imposed a ban on handing over suspects to Afghan forces last November because of fears over ill-treatment.
UK lawyers acting for eight of the men said their clients were arrested by British soldiers in raids in villages in Helmand and Kandahar provinces and have been held for between eight and 14 months without charge.
They claimed it amounted to unlawful detention and internment.
Lawyers for the men, whom the BBC has chosen not to name over fears for their safety, launched habeas corpus applications at the High Court in London on 18 April, with a full hearing due in late July.
Habeas corpus, in this context, argues for the right to be brought before a court to determine whether their detainment is lawful or not.
The lawyers argue that the MoD should release the men because the British army has no power to continue holding them.
A senior government lawyer, James Eadie QC, described this situation in court as a "perfect legal storm" because the Army suspects each detainee of links to insurgents.
The families of two of the men who appear to have been held the longest said they were arrested in the spring of 2012 and interrogated in the weeks that followed.
But legal papers state their interrogation ended "many months ago".
Last week, the two were allowed access to lawyers but they have still not been told why they are being held and they have not been charged with any crime.
The families only established where the men were being held with the help of the International Committee of the Red Cross.
One, a teenager, has been held for 14 months, while the other, a 20-year-old father, has been held for 12 months.
In legal papers Dan Squires, a barrister for the 20-year-old, told the High Court: "He has not been granted access to lawyer nor brought before a court.
"He does not know how long he is to remain detained or for what purpose. He has asked whether he will be transferred to Afghan authorities but had been told they do not consider that he has committed any criminal offence and so do not want to receive him."
In preparatory legal arguments at the High Court on 22 April, Mr Justice Collins told the government that the case raised serious questions about the British army's power to hold suspects in Afghanistan, because the UK could not operate a Guantanamo Bay-style prison - referring to the US facility in which enemy combatants can be held indefinitely without trial.
UK forces operate in Afghanistan as members of Isaf.
Isaf was established by a UN security council resolution which "authorises the member states participating in Isaf to take all necessary measures to fulfil its mandate".
The security council resolution does not include a power to intern detainees on the basis that they are regarded as a threat to national security or otherwise.
The MoD said detainees were held where there was evidence linking them to criminal activity.
"Many are either suspected killers of British troops or known to be involved in the preparation, facilitation or laying of improvised explosive devices. They are held pending transfer to Afghan authorities for further investigation prior to prosecution," a spokesman said.
Phil Shiner, lawyer for eight of the men, said: "This is a secret facility that's been used to unlawfully detain or intern up to 85 Afghans that they've kept secret, that Parliament doesn't know about, that courts previously when they have interrogated issues like detention and internment in Afghanistan have never been told about - completely off the radar.
"It is reminiscent of the public's awakening that there was a Guantanamo Bay. And people will be wondering if these detainees are being treated humanely and in accordance with international law."
In a statement, the MoD confirmed the existence of a temporary holding facility but would not comment on the numbers of detainees held there, nor on individual cases.
It said the detentions were legal under the UN mandate and complied with international obligations.
A spokesman said: "Detention operations are an important part of our force protection measures protecting British troops, our allies and partners, and the Afghan civilian population.
"They directly contribute to the success of the Nato Isaf mission in Afghanistan and ultimately to UK national security. The threat of UK court action is currently preventing us from transferring detainees to the Afghan authorities."