Both Mr Blair and Mr Brown deny they agreed to hold the inquiry in private.
Gordon Brown has denied Conservative claims his stance on whether to hold an inquiry into the Iraq war in public has undergone a "U-turn in slow motion".
The PM had said it would be held in private for security reasons but later said some sessions could be in public.
He said he wanted the inquiry to have all evidence that is necessary" including confidential material.
Meanwhile the Lib Dems say they have been told former PM Tony Blair will be asked to give most evidence in public.
Mr Brown faced widespread criticism of his original announcement to MPs that the inquiry would be held in private from some military families, Lord Butler, former PM Sir John Major and others.
Days later Mr Brown wrote to the inquiry's chairman, Sir John Chilcot, saying it was up to him to decide if some sessions should be held in public.
On Monday Sir John replied saying he felt it was "essential to hold as much of the proceedings of the inquiry as possible in public" consistent with national security and encouraging "complete candour" from witnesses.
Asked if the Tory claim he was "executing a U-turn in slow motion" was right, Mr Brown told BBC Radio 4's World at One: "No, I don't think so."
He said the scope of the inquiry - covering eight years of a "very controversial issue for Britain" - meant it would always be difficult.
But he said he had always said there would be a "process of consultation" with other party leaders and senior MPs.
"I'm trying to find a way of getting an inquiry that can satisfy people that we're doing everything in our power to get to the truth while at the same time I think everybody understands ... you've got to take into account national security considerations and that you've got serving military who will want to give evidence .. sometimes in private."
He said the Conservatives had been asking for a Franks-style inquiry - a reference to the committee that reviewed the Falklands War - which was partly held in private.
He added people who had something to say that was "confidential or effects our relationships with other countries" could say it "directly to Chilcot" but there would be "scope for people to give evidence in public if that is what he chooses".
Meanwhile Liberal Democrat leader Nick Clegg has met Sir John and was told former PM Mr Blair would be asked to give some public evidence, apart from on sensitive issues of national security.
The Conservatives have prompted a Commons debate on the inquiry on Wednesday. Leader David Cameron told the BBC he would be pushing for "some clarity" and hoped to get "a proper U-turn" out of Mr Brown.
"We want to know that the meetings will be held predominantly in public. We want to know that Tony Blair will give evidence in public, that the prime minister will give evidence in public," he said.
"And we want to see an inquiry committee with more military members, with more political experience, [that is] much more heavyweight and diverse."
Downing Street has said the prime minister will co-operate fully with the Iraq inquiry, including giving evidence in public if required.
The Lib Dems said Sir John also said witnesses would not be made to give evidence under oath as it was a non-judicial inquiry, but an equivalent format would be found.
And the party was told he would make up for the gaps on the inquiry panel by asking military experts for help.
Mr Clegg's spokesman said Sir John had made "a sincere attempt to make up for the shortcomings of Gordon Brown's initial announcement".
But he said Mr Clegg wanted guarantees that there would be proper cross examination of witnesses.
Downing Street and Tony Blair's spokesman have dismissed reports that the decision to hold the inquiry in private was prompted by pressure from the ex-prime minister.