Those seriously injured on the day will also be offered compensation.
According to the BBC, the offer was made following months of discussion between lawyers for the Ministry of Defence and the families' legal teams.
The figure has been circulated in an email, but it is not clear yet if a formal letter has been sent to the families.
Lord Saville drew up a landmark report which criticised the Army over the killings.
His panel ruled that the Army fired first and without provocation.
It found all 14 who died and the others who were injured almost four decades ago had been unarmed and were completely innocent.
The troops had also continued to shoot as the protesters fled or lay fatally wounded on the ground. One father was shot as he went to tend to his injured son, the mammoth 5,000-page report revealed.
Soldiers later insisted they had only retaliated, in a bid to cover-up the truth, the document - described as "shocking" by Mr Cameron - said.
"We found no instances where it appeared to us that soldiers either were or might have been justified in firing," it said.
"Despite the contrary evidence given by soldiers, we have concluded that none of them fired in response to attacks or threatened attacks by nail or petrol bombers. No one threw or threatened to throw a nail or petrol bomb at the soldiers on Bloody Sunday."
Bloody Sunday was one of the worst state acts of the conflict and helped ignite 30 years of violence by the IRA.
Victims have spent years campaigning for justice and the revision of an original probe into the massacre which they branded a whitewash.
LIST OF VICTIMS OF BLOODY SUNDAY
Patrick "Paddy" Doherty (31)
Gerald Donaghy (17)
John "Jackie" Duddy (17)
Hugh Gilmour (17)
Michael Kelly (17)
Michael McDaid (20)
Kevin McElhinney (17)
Bernard "Barney" McGuigan (41)
Gerald McKinney (35)
William "Willie" McKinney (26)
William Nash (19)
James "Jim" Wray (22)
John Young (17)
John Johnston (59)
John Johnson was shot twice on January 30, 1972 and died on June 16, 1972. His family say he died prematurely and that his death was due to the injuries received and trauma he underwent on Bloody Sunday.
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