A vote by MPs to ask for a Brexit delay has been welcomed by Leo Varadkar.
The taoiseach (Irish prime minister) said the vote "reduces the likelihood of a cliff edge, no-deal Brexit".
Theresa May said Brexit could be delayed by three months, to 30 June, if MPs back her withdrawal deal in a third vote, but could be delayed for longer if they reject her deal again.
Mr Varadkar urged the UK government to make clear the purpose of an extension "and how long it would last".
The vote in the Commons on Thursday night, means the UK may not now leave on 29 March as previously planned.
Any delay has to be agreed by the other 27 EU member states.
On the EU agreeing a potential delay, Mr Varadkar said: "We need to be open to any request they [the UK] make, listen attentively and be generous in our response."
The taoiseach said a potential further delay would be discussed at a European Council meeting later in March.
Mr Varadkar's remarks follow comments by Ireland's Tánaiste (Deputy Prime Minister) Simon Coveney, who said a Brexit extension of 21 months is possible.
Earlier on Thursday DUP leader Arlene Foster said it was possible a deal on Brexit could still be reached in the coming weeks.
The "final part of a negotiation" is when it matters most, the DUP leader said, as it is "when you start to see the whites in people's eyes".
Her party is in discussions with the government, amid reports it could back the PM's Brexit deal.
This follows MPs' rejection of the idea of leaving the EU with no deal.
Mrs Foster, who is attending St Patrick's Day events in Washington DC this week, said DUP representatives were speaking to the government and Attorney General Geoffrey Cox about changes to the deal.
"Nobody wants to leave without a deal and we want to make sure we get there," she told BBC News NI.
The government's deal was rejected for a second time on Tuesday in Parliament.
The DUP leader said her party wanted the UK to leave the EU with a deal, but that her party had certain tests it must meet before they will back it.
So what is @duponline talking to the government about? One source says "tweaks won't cut it" & that the party is "continually engaged with a range of govt figures to ensure every element of the UK is treated the same"— Mark Devenport (@markdevenport) March 14, 2019
"It's very simple - what it will take to get the DUP over the line is the fact that Northern Ireland is not left behind, the constitutional integrity of the UK is the same and we have a strong say in the future of the UK," she said.
"Brexit is two weeks away, as I've constantly said, when you come to end of a negotiation that's when you really start to see the whites in people's eyes and you get down to the point where you can make a deal."
Analysis: DUP centre stage once more
By Jayne McCormack, BBC News NI Political Reporter
It's no surprise that the DUP find themselves centre stage in the Brexit soap opera once again.
Their votes have been crucial throughout the Brexit process, and they haven't been shy of reminding the PM how much power they wield from time to time.
That being said, Wednesday night's votes in Parliament surely did not go how the DUP had hoped.
And the government's publication of its no-deal tariff plans for Northern Ireland have turned up the political heat.
Quiet conversations will be happening all across Westminster, and in Washington today, to see if there is any way through this cloud of political smog.
Mrs Foster also confirmed that she had discussions with Mr Varadkar in Washington on Wednesday.
She said they had a private meeting where they talked about a range of matters, but that Brexit was on the agenda.