Relatives of Joseph Murphy are requesting the move amid claims the 41-year-old was shot a second time while being held in custody at an army barracks.
It has been alleged that Mr Murphy, who was shot in the thigh but lived for 13 days after the incident in west Belfast, had another bullet fired into his wound.
His family have applied to the Belfast's Coroner Court to exhume his body saying that medical documents of the time had not properly recorded his injuries.
It comes as the preliminary hearing of the inquest into the Ballymurphy shootings took place.
Mr Murphy's daughter Janet Donnelly on Monday said: "My daddy said he was shot into his open wound, he thought it was a plastic bullet, now the family are saying this hasn't been a plastic bullet, I believe it to be a live round - it's the only explanation."
There is only one entrance wound but there's an exit wound, so where did the second bullet come from, so that's what I am asking, that's why I am asking for the body to be lifted
A Catholic priest and a mother-of-eight were among the 10 people shot by paratroopers in Ballymurphy in 1971, while an 11th person died of a heart attack suffered during the carnage.
The Army has claimed that its troops opened fire on the day in question after being shot at by republicans, as it acted to round up suspected paramilitaries after internment was introduced.
However, the families of those killed have always maintained their innocence. Previous calls for an independent investigation have been refused by the Secretary of State.
John Teggart, whose father was shot 14 times by Paratroopers in west Belfast 40 years ago, said he believes anyone who lost a loved one should be entitled to legal redress.
It comes after former Secretary of State Peter Hain defended controversial letters which were sent to 187 'on-the-run' paramilitary suspects, telling them they were no longer wanted.
Mr Teggart told UTV: "There should be no special deals.
I think anybody out there whose parents were murdered in cold blood - whether it's by the army, IRA or anybody - would be entitled to what we are doing
"We are entitled to the legal requirements, with the evidence there which we have gathered, to put these soldiers in the dock, there should be no special deals and that is what the Ballymurphy families have agreed on."
Over the weekend, Peter Hain said pursuing those responsible for crimes during the Troubles "seems a waste" and that Tony Blair's government acted "honourably" over the controversial OTR letters, which were the centre of a huge political row in Northern Ireland last week.
The Labour MP also said it was "risible" for politicians to claim they had no knowledge of the so-called comfort letters, which were sent to those who may be wanted to answer for crimes committed prior to the Good Friday Agreement.
John Teggart continued: "It's been hard, but it makes it hard when you get Mr Hain, an ex-Secretary of State ... we've been through three Secretary Of States since we started the campaign, so what Peter Hain is saying doesn't really matter to us.
"We're not listening to somebody who is an ex-Secretary of State but let's move on to what's happening with the British government that's in power now."
As the families gathered in Belfast ahead of the preliminary hearing of the Ballymurphy inquest, John Teggart said it has been worthwhile carrying on with their campaign.
He said: "It's a good day for the families. For over 40 years the families have been denied justice. There was inquests called that could be called the Ballymuphy 'Widgery' - there was no eyewitnesses called, no civilian eyewitnesses called, and it was a total farce.
"My father was shot 14 times, there was a case there to be answered. We didn't get any justice then and this is the start of a legal process which we were entitled to 40 years ago."