Coveney said the government is still warning against all non-essential travel, but that a green list of countries “where it will be safe to fly to and from“ will be published by the government next week.
Speaking at a post-cabinet briefing in Dublin on Monday, he said: "People should not be coming to Ireland if they can't restrict their movements for 14 days when they come here.
"This is not a time for normal holidays for tourists coming from abroad to Ireland. We have been very clear on that.
"There was certainly some evidence over the weekend that there were tourists in Ireland who said that they weren't self-quarantining or restricting movement, and that has created a concern across the tourism industry.
"Irish people who are holidaying in counties like Kerry want to know they can go and holiday with their family safely. We are considering, as a government, the measures we can take to improve communications, to improve protocols in our airports, to ensure that the passenger locator form that passengers who come into Ireland fill out that we can move it online."
He added: "I don't think we should prevent flights from landing in Ireland or ban international travel. You could argue Ireland has the most restrictive system in the EU right now in terms of international travel."
Ireland's acting chief medical officer said mandatory quarantine for travellers arriving into Ireland from abroad would be "desirable", but that it is up to the government.
Speaking at a Department of Health press briefing on Monday, Ronan Glynn said: "From a public health perspective, mandatory quarantine would clearly be a desirable measure but there are wider implications and wider considerations for government in decision making around the issue of mandatory quarantine.
"From our perspective, anyone coming into the country should be coming into this country should be restricting their movements.
"I don't think it is reasonable in the context of a pandemic that could go on for several months to say we can shut down travel completely. From a public health perspective, we want to stop as much if not all non-essential travel if at all possible.
"If there is a small number of countries identified that has a similar profile in terms of the virus to ourselves, then from a public health perspective we wouldn't be concerned about that small list of countries.
No new coronavirus-related death have been reported in Ireland on Monday, leaving the total at 1,746, the National Public Health Emergency Team said. ■
Under an intense surge of arctic air, Friday morning will begin with the coldest temperatures so far this season across much of the central and eastern U.S. with blustery conditions and a piercing wind chill.