The former motor racing boss said he inherited the money from Oswald, adding his father received it from his father with the cash passed down through the years.
Mr Mosley said the cash for the only officially-recognised UK press regulator is “family money” from a family trust, adding it is “impossible” for him to exert any influence over Impress.
He also said it is “pure guff” for media outlets to suggest the proposals for press regulation would benefit conmen, warlords and politicians.
The Independent Press Regulation Trust agreed to a £3.8 million four-year funding deal for Impress and said this money has been guaranteed by the Alexander Mosley Charitable Trust.
This was set up by Mr Mosley, a victim of a newspaper sting involving images of him at a sex party, with Impress also receiving a “generous” donation from writer JK Rowling.
Newspapers could soon be forced to pay their opponents’ legal costs linked to libel and privacy actions, even if they win in court, if they are not signed up to an officially-recognised regulator.
Section 40 of the Crime and Courts Act 2013, which contains the provisions, has yet to be implemented - much to the frustration of victims of press abuse.
Questioned about where the money for Impress has come from, Mr Mosley told BBC One’s Sunday Politics: “The money comes from a family trust. It’s family money. The thing is you have to understand somebody had to do this.”
Pressed on the issue, and whether it was put together by his father, Mr Mosley replied: “Not put together by my father - my father inherited it from his father and from his father.
“The whole of the middle of Manchester once belonged to the family, that’s why there’s a Mosley Street.
“This is actually completely irrelevant because as we have given the money, I have absolutely no control.
“If you do the most elementary checks into the contract between my family trust, the trust that finances Impress, it is impossible for me to exert any influence. It’s just the same as if it’d come from the National Lottery.”
Presenter Andrew Neil said people would find it ironic that money to fund the press regulator has come from Britain’s “historically best known fascist”.
Mr Mosley replied: “You see it hasn’t.”
He added it has come from a family with deeper historical roots than his father, adding: “That’s why there’s a Mosley Street in Manchester. This is complete drivel because the fact of the matter is we have no control.
“Where the money comes from doesn’t matter. If it had come from the National Lottery, exactly the same, Impress is completely independent.”
Mr Mosley also told Mr Neil: “I do not have a vendetta of the press and stop saying that, it’s not true. You submit evidence for it or don’t put it forward.”
Mr Mosley repeatedly said the press could start their own Leveson-compliant regulator, stating this would leave them with no complaints.
Most newspapers have signed up to rival regulator the Independent Press Standards Organisation (Ipso), the press-funded body which has not sought official recognition.
Ms Mosley said: “What they all omit is a full reading of section 40 because that cost shifting will only apply if ‘it’s just and equitable in all circumstances’.”
Mr Neil asked Mr Mosley to address the view that media outlets think the proposals are a “charter for conmen, warlords, crime bosses, dodgy politicians, celebrities with a grievance against the press”.
Mr Mosley replied: “It’s pure guff and the reason they’re doing this is they want to go on marking their own homework, as they have ever since the war, despite seven investigations.
“So the fact of the matter is the press do not want to make sure that life is fair.
“All I want is somebody who has got no money to be able to sue in just the same way as I can.”
Conservative former culture secretary John Whittingdale said he did not believe the Government will repeal section 40.
He told the same programme: “What I’m arguing for is not to implement it but it will remain on the statute book.
“And if it then became apparent Ipso simply was failing to work, not delivering effective regulation and the press were behaving in a way that was wholly unacceptable as they were 10 years ago, then there might be an argument at that time to think in that case we are going to have to take further measures, of which section 40 might be one.”