Garda Commissioner Nóirín O'Sullivan has insisted she neither authorised nor was aware of any attempt by the force to target a garda whistleblower.
In a statement this afternoon, An Garda Síochána said it wanted recent allegations examined as soon as possible.
The statement said: "Commissioner O'Sullivan would like to make it clear that she was not privy to nor approved of any action designed to target any Garda employee who may have made a protected disclosure and would condemn any such action."
The commissioner said she wanted to reiterate that any whistleblower with concerns would be taken seriously and have the matters raised examined.
In the Dáil this morning the Taoiseach raised the prospect of a judge-led inquiry to examine protected disclosures made by two gardaí who are said to have claimed a campaign to discredit a whistleblower had been sanctioned by An Garda Síochána leadership.
The Government yesterday confirmed that Tánaiste and Minister for Justice Frances Fitzgerald received allegations that senior gardaí were involved in a campaign to discredit a whistleblower in the force.
The minister told the Oireachtas Committee on Justice today that she was assessing the body of evidence in relation to the allegations.
She said she was analysing the best possible route to deal with the evidence she received earlier this week under the Protected Disclosures Act 2014.
She said her main consideration was to ensure that the rule of law is applied to those allegations and that a "just and real" procedure is put in place.
Ms Fitzgerald said her job was to follow the letter of the law and she assured members of the committee she would do that.
In the Dáil today, Sinn Féin deputy leader Mary Lou McDonald asked whether Ms Fitzgerald and Commissioner O'Sullivan were running for cover.
Independents4Change TD Clare Daly said Ms O'Sullivan's position is untenable and that she "should go".
Mr Kenny acknowledged that the claims were very serious. He said it would be utterly unacceptable if whistleblowers were not treated properly and his Government would not shirk its responsibility in this regard.
The Taoiseach said the allegations would have to be examined and a sitting judge might have to be appointed to do that.
But Ms McDonald said "the revelations" were contributing to declining morale in the force and would cause future whistleblowers to think twice before coming forward.
She said nothing had changed since Sgt Maurice McCabe and Garda John Wilson had been smeared.
Ms Daly claimed a whistleblower had been subject to a campaign to "discredit and annihilate" him on the watch of the current commissioner and she said that it is time for the commissioner to go.
But Mr Kenny said he had full confidence in Commissioner O'Sullivan and Minister Fitzgerald.
Speaking to RTÉ's Morning Ireland and later in the Dáil, Ms Daly said she knew "for a fact" that people who have come forward and made protective disclosures on the watch of Commissioner O'Sullivan have had no contact with her, but that people in the stations who bullied them have been included on promotions list.
She added that no garda could be confident in coming forward, given the attitude towards whistleblowers.
"I think that the commissioner's position is utterly untenable," said Ms Daly on Morning Ireland.
"There's a huge breach of trust now in this regard. How could any garda have confidence in that operation and certainly seeing what these whistleblowers have been through. I wouldn't if I was one."
Ms Daly said that transforming the top tier of An Garda Síochána was unsuccessful because Commissioner O'Sullivan had worked closely with former commissioner Martin Callinan and there had not been a cultural change within the force.
Fianna Fáil's justice spokesman, meanwhile, has called for a prompt and thorough investigation.
Speaking on RTÉ’s News At One, Jim O'Callaghan said: "What we need to do is set up a proper mechanism for them to be investigated immediately.
"It's extremely serious the allegations that were made in the Examiner article yesterday and that has to be fully and adequately investigated.
"And there are a number of methods going down that route - we can either appoint a senior barrister or a judge for the purpose of investigating the allegations, alternatively, they are so serious that they may merit asking Mr Justice O'Higgins to continue in his investigations and to examine the specific allegations, but they must be investigated, thoroughly."
Mr O'Callaghan said he does not have sufficient information at present to make a suggestion that anyone should resign.
"I'm not the kind of politician calling for people to resign unless I see coherent evidence that they have been engaged in wrongdoing.
"If the investigation shows that the Commissioner was aware of it or through inadvertence or negligence wasn't aware of it, she'll have questions to answer, but I'm not prepared to start calling on people to resign when we don't have findings. I don't jump to conclusions until I see a final report on foot of an investigation."