The Taoiseach has said that he does not agree with those who say there should be a Brexit minister.
Mr Kenny told reporters that he chaired the Cabinet committee on Brexit and every minister was in effect a Brexit minister.
In a briefing to journalists, he said it was far too serious a matter to be hived off to a section of a department or to a minister who had other responsibilities.
Mr Kenny said Brexit preparations would intensify over the next couple of weeks, adding that there will be an all-island forum in Dublin next Wednesday and sectoral meetings around the country.
He said he will be travelling to the border counties next week and there will be an important North-South ministerial meeting in Armagh in November.
The European Council meets in December when the 27 leaders will discuss Brexit without British Prime Minister Theresa May.
Mr Kenny also said he will be leading a trade mission to the US to emphasise that Ireland continued to be an attractive place to invest.
Meanwhile, the Minister for Agriculture has described as "naïve" a suggestion that a "Minister for Brexit" would solve the problems that have arisen since the 23 June referendum and Britain’s decision to leave the European Union.
Michael Creed told the Dáil that Brexit is a "whole Government issue".
Responding to Rural Alliance TD Mattie McGrath, Mr Creed said the significance and importance of the matter was indicative by the fact that it is "run" out of the Department of the Taoiseach.
It is understood the DUP has not yet responded to an invitation to attend the all-island forum on Brexit next week.
The North's First Minister and DUP Leader Arlene Foster previously shot down a proposal from the Government for such a forum at a meeting of the North South Ministerial council in Dublin last July.
Separately, it is understood that the Irish authorities do not believe British immigration officials will be based in Ireland post-Brexit as a means of strengthening external border of the common travel area.
Investors exploring option of moving to Ireland post-Brexit vote
It is understood that since the UK voted for Brexit, the IDA has received queries from over 100 investors about moving part or all their operations here.
The most advanced project would involve several hundred jobs.
Half of the queries relate to companies in the financial services. The other queries relate to technology, internet and manufacturing companies with a supply chain that extend into the EU such as pharmaceutical companies.
It is believed that some of the queries are now translating into site visits with the arrival of technical teams to Ireland.
The queries relate to areas across the country but it is understood there is a bias to locations in the greater Dublin area.
Meetings are taking place in relation to some of the queries in Britain and in the corporate headquarters of the companies.
It is too early to say how many of the queries will result in companies relocating in some manner here and providing additional jobs.
It is understood some companies will restructure their operations post-Brexit here but London will still be a major financial centre.
Separately an Irish official working in the European Commission has been appointed to the Commission's Brexit taskforce, which is headed up by the Michel Barnier.
It is believed the Irish authorities had sought to have a person with an Irish background appointed as they would be familiar with Irish concerns. However they will not be doing Ireland's bidding as is the situation with the other officials on the taskforce.