Minister Charlie McConalogue insisted that high Covid-19 infection rates in the north-west are “not a Border issue”.
His remarks came as he was pressed on whether there should be enforcement measures to prevent cross border travel.
Northern Ireland's Health Minister Robin Swann has written to the Minister for Health in the Republic Stephen Donnelly saying non-essential cross-Border travel must be stopped "by enforcement if required".
In a letter to Mr Donnelly on Wednesday, Mr Swann warned of a “fresh increase of community transmission of Covid-19”.
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He said that governments in both jurisdictions should do everything possible to prevent non-essential cross-Border travel.
Ministers in the south have given no indication that the Government intends to prevent non-essential cross-Border travel as a result.
On Saturday Fianna Fáil junior minister Thomas Byrne told RTÉ: "We are opening up inter-county travel on Monday. That doesn't mean that everyone has to go gangbusters at it."
He added: “But if Robin Swann feels the need to introduce restrictions on the North that’s a matter for them. But we haven’t done this on a 26-county basis at any point it’s always been done the 32-county basis.”
Mr McConalogue, the Minister for Agriculture and a Donegal TD welcomed what he said was Mr Swann's willingness to work on a cross-Border basis in dealing with the virus.
He said: "We have seen, for example in Donegal here, a very close correlation with Derry and Strabane in relation to infections over the last number of months" but also highlighted the rollout of vaccines on both sides of the Border.
He was asked on the BBC’s Sunday Politics programme if there should be enforcement on the Border.
Mr McConalogue said: “I think the key point here is that it’s not a Border issue, it’s a regional issue, and we have to work together as we gradually reopen.”
He also said: “From [Monday] we’re in the situation because of the increasing rollout of vaccination to be able to allow intercounty, non-essential travel.
"And that's because we've got into that space and likewise that is the situation in Northern Ireland.
“So I think it’s really good that people can now travel between counties for non-essential purposes but in doing that our clear message to everyone is to continue to stay safe and continue to be very aware of the possibility of infection if we let our guard down.”
Dr Tom Black, the chairman of the British Medical Association in Northern Ireland told the same programme he does not believe enforcement is necessary at this point.
He said more people have been vaccinated in the Republic than in the North and the programmes are working well to keep transmission of the infection down in older people.
Dr Black said the vaccination programme is “working really well” to keep both transmission and infections down among older people “who would end up in hospital”.
The “granular detail” of the data suggests mostly under-40s are getting infected. This age group mostly don’t end up in hospital and usually don’t end up very sick, he said.
While there would be a risk of bringing the infection back to older age groups, they are “generally speaking now vaccinated” in the North, he said.
Some 98 per cent of over-60s have been vaccinated in the North.
“It is very clear what is happening in the north west,” said Dr Black.
“Young people are congregating in the back garden to have a few drinks. It gets very cold and they go inside and then half of them catch the infection from the one person who brought it in.”
Dr Black said what is needed is for young people to be effectively warned “you can’t be mixing inside, you can’t be having drinks and congregating and mixing and then taking [the virus] home”.The Derry-based GP said “localised outbreaks” were always expected while test-and-trace programmes are needed on both sides of the Border.
“This is to some extent what we expected. It has not got to a crisis yet.”
