The Taoiseach says it is "too early" for British tourists to come to Ireland without quarantining for two weeks.
Micheál Martin said that due to a rising reproduction rate, and noting the UK's "difficulties" with suppressing the virus.
Concern has been flagged in recent days after reports that British tourists were coming to the state and not following the travel advice.
On Saturday, a further 820 cases were reported in the UK, the highest number for ten days.
Micheál Martin says Ireland is being "very cautious" on travel generally.
"We would be very cautious on international travel, generally," he told the BBC's Andrew Marr show.
"On the 20th of July, 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 relationship with other countries, in relation to the level of the disease in particular countries, including the UK.
"In terms of the level, we will be looking at countries that are either on the level or below, that is the metric that I go with.
Quarantine 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 the numbers. We're very concerned about that, and the cabinet will be discussing this again this week, in terms of measures that we may need to take to spend our resources at airports and so on to prevent the spread of this virus so it's a serious issue for us we're taking a cautious approach.
"Suppression of the viruses is tricky. We watched reports where in certain parts of the UK there is difficulties where certain areas are not down.
"Our travel advice is not to travel for non-essential purposes, that's our advice, because our priorities to get our schools open towards the end of August to also free up hospital capacity to deal with non-Covid illnesses and to try and get more activity there.
Caution is the watchword here.
Mr Martin also told Mr Marr that Brexit talks needed "an injection of momentum".
"I had a very crucial discussion with the Prime Minister Boris Johnson, after I took office, I think we both agreed that it's in everybody's interest, particularly in terms of businesses and giving them certainty and securing employment into the future that we can include comprehensive trading between the UK and European Union, that can be tariff free quota free, and that will facilitate smooth future relationships and good access to the single market.
"So far, progress has been too slow, in my opinion, and we do need an injection of momentum here, and we do need to see a sense that people want to bring these discussions to a conclusion, and to have a timeline for the conclusion of these discussions."