British Prime Minister Rishi Sunak has been formally asked to apologise for the treatment received by the Hooded Men almost 50 years ago.
Lawyers acting for some of the men have written to Mr Sunak after the PSNI apologised to the majority of survivors on Tuesday.
One of the men, Joe Clarke, who died on Monday, received a death-bed apology last week.
The men say they were subjected to state-sanctioned torture when they were held without charge in 1971.
In 2021, the Supreme Court ruled that the methods used "would be characterised today" as torture.
The techniques used against the 14 men included being hooded, made to stand in stress positions, forced to listen to loud static noise and being deprived of sleep, food and water.
In its apology the PSNI recognised the ill treatment the men were subjected to more than half a century ago.
It said: "The Police Service of Northern Ireland acknowledges the finding of the United Kingdom Supreme Court that it is likely that the treatment to which you and the other Hooded Men were subjected to at the hands of the security forces, including some police officers, would be characterised today as torture.
"We wish to acknowledge that the treatment you received was not acceptable at that time and is not acceptable by modern standards of policing.
"We would like to convey an apology to you for the actions and omissions of police officers at that time."
Solicitor Darragh Mackin, of Phoenix Law, was involved in negotiating the apology over the past 18 months.
He has now written to Downing Street, the Secretary of State and Ministry of Defence.
"We write formally to you today to respectfully request that you now, to use the PSNI’s phrase, do 'the right thing' and apologise," Mr Mackin wrote.
"The fact that the PSNI, the devolved policing authority in this jurisdiction, have taken such an unprecedented step ought not to be disregarded."
In response to Tuesday's development the British government and MoD issued identically worded statements.
It read: "We acknowledge the pain and suffering felt by so many during the Troubles.
"There are several ongoing legal proceedings in relation to this incident, and therefore it would be inappropriate to comment further at this stage."
Mr Mackin raised concerns about the reference to current legal proceedings.
"Such a reference is misplaced and indeed, fundamentally erroneous," he wrote.
"Such a bland assertion fails to acknowledge the very fact that your criminal justice partner (PSNI) in this jurisdiction who is in an identical position, has deemed it appropriate to proceed as outlined above."
Mr Mackin also called for the assistance of Irish foreign affairs minister Micheál Martin "in engaging with his respective counterparts in the UK Government".
"The purpose of this request is to seek a formal acknowledgement of the actions of the UK Government and the issuing of a formal apology to ‘the Hooded Men’ without any further delay," he wrote.
The NIO, MoD and Department of Foreign Affairs in Dublin were contacted.
Meanwhile, Mr Clarke's funeral is due to take place on Thursday during which surviving members of the Hooded Men will form a guard of honour.
His remains are due to leave his home at Brackenwood Lane in south Belfast before they are brought to Healy Brothers funeral home on the Andersonstown Road, in west Belfast.
It is expected the national flag will then be placed on his coffin, which will be flanked by a guard of honour.
His remains will then be walked to Milltown Cemetery where Mr Clarke's close friend Jim McIlmurray will deliver an oration.
Finally, members of the Hooded Men group will form a final guard of honour as his remains are lowered.
Mr McIlmurray said Mr Clarke planned his funeral several weeks ago.
"I planned this with Joe three weeks ago," he said.
"He wanted it to be a celebration of his life, he did not want it to be a mournful event.
"He said 'you will do me proud'."
Mr McIlmurray said he was "very honoured" to be delivering the oration but "somewhat humbled".
He said Mr Clarke wanted his old friends to send him off.
"He said 'I want my former comrades to stand at the graveside as I go down'," Mr McIlmurray said.
"I was with them to the end and they will be with me to my end."