The time available to reach a Brexit deal is "shrinking rapidly", according to the Government.
It comes as the European Commission President Jean-Claude Juncker has warned a hard border would return in Ireland if there is a no-deal Brexit.
Taoiseach Leo Varadkar is heading to New York, where he's due to meet British Prime Minister Boris Johnson in the coming days.
While the two men are in the US for the UN General Assembly and climate change summits, they will "take stock" of Brexit negotiations during their meeting.
In a statement, a spokesperson for the Government said Mr Varadkar will reiterate the Ireland is open to considering any "detailed, legally operable and realistic" alternatives to the Irish backstop.
However, it also stresses that "significant" gaps remain between the current ideas being put forward by Boris Johnson's government and the "legal certainty" of the withdrawal agreement.
The statement adds: "Talks are underway between the UK and the EU Task Force and there is a desire on all sides to reach a satisfactory agreement. The EU has made clear to the UK that it is open to examining specific proposals.
"However, as October 31st is just a few weeks away, the time available to reach an agreement is shrinking rapidly and engagements from the UK need to step up a gear in order to reach a deal."
Mr Varadkar is also due to meet with European Council President Donald Tusk during the New York trip.
"We did not invent Brexit"
Meanwhile, Mr Juncker has warned there would be a border between Northern Ireland and the Republic if the UK leaves the EU without a deal.
Speaking to Sky News' Sophie Ridge, he said it would be up to the British government to explain the specifics of how that would work.
He argued: "We have to make sure that the interests of the European Union and of the internal market will be preserved.
"The EU is in no way responsible for any kind of consequences entailed by Brexit.
"The European Union is not leaving the United Kingdom - the United Kingdom is leaving the European Union... We did not invent Brexit, and we were never pleading in favour of any kind of Brexit. It's a British decision."
He also suggested that some people in Westminster are "forgetting about the history" in Ireland.
Mr Juncker claimed "history will be back immediately" if there is a hard Brexit.