Taoiseach Micheál Martin said the Government remains "cautious" on international travel, as its priority is on opening up parts of society safely.
Speaking on the BBC's Andrew Marr show on Sunday, Mr Martin said there is still a lot of volatility relating to Covid-19. He said the Cabinet will be discussing measures this week that may be needed, such as strengthening measures at airports, ahead of issuing a possible 'green list' of countries.
Mr Martin said it is too early for holidaymakers to return to Ireland without having to quarantine on arrival.
When asked if Britain can get on Ireland's green list, he responded that "suppression of the virus is key".
He said the Government has watched reports of where parts of UK have had to go back into lockdown, saying so many countries are facing such challenges and that is why "we are saying not to travel for non-essential purposes".
The focus for the Government here is about opening up society safely, he added.
"Our priority is to get schools open and free up hospitals to deal with non-Covid and get more activity. I think caution is the watch word here and that will be our approach."
He said the Government is developing a methodology similar to the EU and it is taking into account the level of the virus in other countries.
"That will be the metric in advising Irish citizens travelling abroad."
He said the Government is concerned by a possible second wave of the virus given what is happening globally.
He added: "On July 20 we will announce our measures in relation to international travel and essentially we have developed a methodology somewhat similar to the European Union and its relation with third countries; essentially, the methodology will be in relation to the level of the disease in particular countries, including the UK, countries that are at Ireland's level or below.
"In terms of people coming in to Ireland, the advisory and the quarantining still remains and it's under constant review. Why? Because there's a lot of international volatility with this virus, we've seen a spike in numbers. We're very concerned about that."