Sinn Féin leader Mr Gerry Adams and the party's chief negotiator, Mr Martin McGuinness, are members of the IRA army council, the Minister for Justice, Mr McDowell, claimed yesterday.
He claimed that Sinn Féin TD for Kerry North, Mr Martin Ferris, was also part of the IRA's leadership.
In an interview on Today FM Mr McDowell said Sinn Féin and the IRA were "directed by the same leadership". Asked whom he meant, he said: "We're talking about Martin McGuinness, Gerry Adams, Martin Ferris and others."
Speaking shortly afterwards on RTÉ radio, Mr McGuinness denied Mr McDowell's claim. "I reject it absolutely," Mr McGuinness said. "What he has alleged is totally and absolutely false."
Mr McDowell outlined his belief that the provisional movement was "engaged in massive criminality", accusing them of fraud, murder and punishment beatings.
He said it was earning millions of pounds from criminal activity, which it was laundering through a network that included solicitors, accountants and other professionals.
"Many people are sucked into it, some wittingly and some unwittingly," he said.
He said the money had been invested in a series of front companies, most recently communications firms. "They are very interested in local newspapers," he claimed. He also indicated his belief that some of the money was being used to finance Sinn Féin's political operations and election campaigns.
"If [other parties in the Republic] had £10 million or £20 million, there'd be no stopping us, no stopping us at all."
Reacting to Mr McDowell's comments, Mr McGuinness said: "I was a member of the IRA many many years ago, and I outlined all of that to the Bloody Sunday tribunal.
"I am not a member of the IRA, I am not a member of the IRA army council. I am one of the leaders of Sinn Féin."
Asked about Mr Adams and Mr Ferris, he said: "They're not members of the IRA army council."
He also accused Mr McDowell of being hostile to the peace process. "People like Michael McDowell have decided that a window of opportunity exists to attack our party, to criminalise our party in the best tradition of Margaret Thatcher."
Meanwhile, Mr Adams said republicans could not engage in criminality of any sort and that those who did had to be expelled from the movement.
He said he understood the seriousness of the political situation following last week's arrests and seizures of cash, the murder of Robert McCartney and the Northern Bank robbery.
He told a rally commemorating three IRA members in Strabane, Co Tyrone, that his party would "weather the storm" whipped up by political opponents. "Do they think we are not serious about our republicanism? Do they think we are not determined to pursue our legitimate goals? Do they believe our commitment to the peace process is a pretence?"
He said the movement would withstand what he called the "criminalisation slur" and would continue to build political support in forthcoming elections.
"Let me restate here at the memorial to these IRA volunteers that the objective of all republicans must be to create the conditions in which the IRA ceases to be. That is a huge challenge for republicans."