The families had feared that the publication of the 5,000 page report, which took 12 years and cost £200m, could be delayed until the forthcoming general election was over.
But after meeting relatives in Derry with Deputy First Minister Martin McGuinness, Mr Woodward reassured them he would attempt to make it available within days of him receiving it at the end of the month.
"We will do it as quickly as possible," he said on Thursday night. "Hopefully within a matter of days."
Apology
Shortly after, the Alliance leader David Ford, who is tipped to become Northern Ireland's justice minister, apologised for offending the Bloody Sunday families.
Mr Ford has come under fire after a private four-month-old email, in which he described the Saville Inquiry as "pointless", was leaked by the SDLP.
Mr Ford also met relatives in Derry and acknowledged that his email had caused offence.
"This comment was taken out of the context of a hastily-written email and is not a full representation of my views," he said.
"I realise that my choice of language was clumsy and inarticulate, and its publication has caused hurt.
"I am very pleased to have been able to meet with some of the families and I have apologised to them for the comment that caused offence.
Lord Saville's inquiry was set up in 1998 to re-examine the events of January 1972, when soldiers from the Parachute Regiment opened fire on civil rights demonstrators in the Bogside area of Derry, killing 14 people.
It was established after the original Widgery Inquiry in the wake of the killings was dismissed by the families as a whitewash.
"I have always believed that what happened on Bloody Sunday was the unlawful use of force against innocent civilians, and that the Widgery Inquiry was a whitewash," Mr Ford said.
But he maintained he had concerns about the process and cost of the inquiry.
"I continue to have concerns about the process and cost-effectiveness of the Saville Inquiry; however, I share the families' hopes that it will provide both justice".
"I don't think I'm the only person in this society who has serious concerns about spending a couple of hundred million pounds to enrich a small group of lawyers, take a decade over doing it and probably in the end not find anything," he told UTV earlier on Thursday.
Mr Ford's remark has been strongly criticised by relatives of some of the Bloody Sunday victims.
"He undermines what Tony Blair did in setting it up. He undermines Lord Saville and his two colleagues and all the years we have spent at the inquiry and now he's justice minister-elect and I would question if he is fit to do the job because the man lacks total objectivity", Kate Nash, whose brother was killed in 1972, told UTV.
SDLP Foyle MP Mark Durkan also questioned David Ford's credibility.
"I think the families will know when they are talking to him that this is the man who is prepared to be the Minister for standing on his head and turning himself inside out just so that he can be the Justice minister", he told UTV.
The First and Deputy First Ministers had different takes on the controversy.
"Myself and many others in the city of Derry will be considerably offended that David Ford, a leader of a party in the assembly, is of a view the Bloody Sunday tribunal, which means so much to the citizens of Derry but particularly to the families, was pointless," Mr McGuinness told UTV before the evening meetings took place.
"If somebody came along to me and they said: 'Look this has cost around £200m. Is there something better you could have done in Northern Ireland with £200m?' The answer is yes," Peter Robinson said.