The Taoiseach has said the Government will meet tomorrow when there would be further discussions around travel, and he said they would also consider advice from the National Public Health and Emergency Team (NPHET).
Micheál Martin said this would culminate in decisions on 20 July in terms of drawing up a 'green list' of countries.
It is expected that a 'green list' of countries where Irish people can travel to without restrictions will be published next week.
Minister for Foreign Affairs Simon Coveney said the Government is going to upgrade their approach to international travel.
When asked if quarantining when coming into Ireland should be mandatory, Mr Coveney said that of course some people will not restrict their movements and that measures will be announced by the Government in the coming days.
Speaking on RTE's Prime Time, he said: "We are going to tighten up measures around filling out passenger locator forms."
He stressed that it is already mandatory to fill out the form "truthfully", if it is not you can be fined or imprisoned, he said.
"We're not relying on people just to be honest, we are following up on people," he said, "we will have a green list by Monday hopefully."
"On the back of a green list we will be able to provide information to people that they will be safe to travel to them that they are as safe as Ireland," he added.
NPHET met today to discuss a range of issues, including the easing of Phase 4 of Covid-19 restrictions which is due to take place next Monday.
It has expressed concern to Government about the scheduled reopening of all bars next week.
Speaking in the Dáil, Mr Martin said there were concerns in terms of increases in the number of Covid-19 cases over the last two weeks.
He said that house parties were an issue and the Government needed to be strong with their messaging around travel with the view of keeping the rate of Covid-19 cases down.
Responding to a question about Special Needs Assistants, he said he will speak to the Minister for Education about including SNA representative voices in the Oireachtas Special Committee on Covid-19 Response.
The Taoiseach said the objective of the Government is to get a full return to school towards the end of August.
However, he warned any slippage in social distancing requirements could undermine efforts to get schools fully open.
He was responding to Independent TD Peter Fitzpatrick, who said he had been contacted by a large number of SNAs, who had expressed their "deep disappointment and despair" that they were excluded from the committee.
Mr Fitzpatrick also asked what action the Government intended to take to ensure that visitors from the US obey the restrictions that are in place.
Mr Martin said information from the Department of Transport is that the number of people coming in from the US is low.
We need your consent to load this rte-player contentWe use rte-player to manage extra content that can set cookies on your device and collect data about your activity. Please review their details and accept them to load the content.Manage Preferences
Mandatory testing for all passengers 'not practicable'
Earlier, the Tánaiste said that he has not ruled on mandatory testing for people travelling to Ireland from countries who are seen as high risk for the coronavirus.
Leo Varadkar said he is not sure if mandatory testing is practicable, but he does think that Ireland needs to consider, as other countries have done, testing of people who are not on the 'green list'.
He said the plan is to produce a 'green list' of countries that people can travel to without restrictions, "because you are no more likely to get the virus there than here".
However, he said, for countries not on the 'green list' or 'A list', the Government is "examining the possibility of bringing in random testing or testing in some cases as other countries have done because we know that mandatory quarantine is not possible in Ireland".
He said mandatory quarantine would not be legally sound and "we've seen for example in Australia it worked out very badly where the hotels where people have been detained have turned out to be clusters or sources of infections themselves".
Acting Chief Medical Officer Ronan Glynn said that if a small number of countries with a similar profile to Ireland are identified, then the National Public Health Emergency Team would not be unduly concerned from a public health perspective.
He said the team is more concerned with travel to and from countries not on that list.
Sinn Féin leader Mary Lou McDonald said she believes mandatory quarantine for those returning home or arriving in Ireland from aboard is a necessary precaution to protect the island.
Speaking on RTÉ's Today with Sarah McInerney, Ms McDonald said that "every precaution necessary needs to be taken" and the Government needs to look at every option.
She said we can't "simply leave our island exposed to the extent that people can land, people can be asked or requested to quarantine for 14 days and that we just take it on a wing and a prayer that that's happening.
"That's clearly not an acceptable situation."
Meanwhile, Professor Sam McConkey has said people who flout Covid-19 quarantine measures should be jailed .
Speaking on the same programme, the Associate Professor of International Health and Tropical Medicine at the Royal College of Surgeons said it was preferable that people consent voluntarily, but he said the laws in place since 28 May provide for fines and imprisonment for those who are found not to have completed contact tracing forms accurately.
"Negotiation or threat should come first, but we really should implement what we said we would do. Our public health is threatened," Prof McConkey said.
He also said he believes continued testing for Covid-19 should be carried out on those who arrive in Ireland.
Additional reporting Fergal Bowers
