The Minister for Justice has urged any person with credible information of corrupt payments to a serving Minister to bring that information to the attention of the Garda Commissioner.
Mr McDowell said there was speculation in the media that a Fianna Fáil Minister in the present Cabinet had received bribes of £80,000 from a developer but he was unaware of any such story being true.
"Nobody has been willing to say in the media who that member of the Cabinet is, nobody has been willing to say to me privately that they suspect any individual of being the recipient of those bribes. I'm wholly unaware of any such story being true, and I'm wholly unaware of any credible evidence that it's true," Mr McDowell said.
"Mary Harney has been assured by the Taoiseach that, as far as he knows, there is no truth whatsoever in that claim and that's the position. We have to govern. We can't deal with every rumour that the media bounce between each other on a ping-pong basis and respond to it.
"But I want to say this, in particular to members of the media who are in possession of credible evidence, if they think they have it that any member of the present Government is or has acted corruptly, go to the Commissioner of An Garda Síochána, go to any tribunal to whose terms of reference such information is relevant, come to any member of the Government with that credible information if you think the Taoiseach, Tánaiste or any other Minister should act on that information.
"Put it in our hands, but don't go on with this ridiculous feeding frenzy in which one journalist publishes something and then every other journalist demands that we should respond to it when nobody in the media is willing to put any of the facts before me so that I can form any judgment as to whether there is any basis to it."
Mr McDowell, who spoke to the media before attending a function at a Dublin hotel yesterday, appealed to journalists to "get a grip on themselves".
"You are in a feeding frenzy at the moment and in fact mutual hysteria - one columnist reading what another columnist is writing - is creating an air of total unreality."
Referring to the Flood tribunal report and its labelling of former Fianna Fáil minister Mr Ray Burke as corrupt, he insisted the Progressive Democrats would not be pulling out of Government over the appointment of Mr Burke to Cabinet five years ago.
"The Flood tribunal has now reported. It has found that Mr Burke was corrupt. Now I don't know what people want to be done now. Is it the media agenda that the PDs should now withdraw from Government in the year 2002 because Mr Burke lied to the Taoiseach in 1997? Is there any credibility to that or do we get on to discuss the real problems which confront this country at the moment, problems to do with the economy, problems to do with jobs, problems to do with Nice.
"I'm second to none in turning my face against corruption. I regard Mr Burke as a disgrace. I reject completely all the explanations he tendered.
"I abhor all the lying that he did in public, but the question we now have to face is who governs this country and what do we do with government, and the media are not going to force the Progressive Democrats into some kind of crisis over what happened at the beginning of the last government when there has been in the meantime a tribunal which has established the truth, a tribunal established by the Progressive Democrats and Fianna Fáil."
However, he said that at the time the 1997 government was formed, there were doubts over Mr Burke. "They were investigated. It was an inadequate investigation. . . he lied at the first stage when he was the subject of a Taoiseach's inquiry, he lied subsequently when he spoke to Dáil Éireann about these matters."