Taoiseach Leo Varadkar spoke with British prime minister Theresa May about the outcome of the Chequers cabinet by phone on Saturday afternoon.
The leaders were speaking after Ms May persuaded her cabinet to back a soft Brexit.
Ms May briefed the Taoiseach on the outcome of her cabinet meeting on Friday.
A statement issued on Friday at the end of a day-long meeting at Chequers suggested that Britain was ready to reach an early agreement on the so-called backstop to guarantee that the Border remains open.
This would maintain full regulatory alignment with Europe for goods and agricultural products, and treating Britain and the EU as a single customs territory.
The Taoiseach welcomed the fact that the UK Government had reached a position where it would put forward detailed proposals for the future EU-UK relationship post-Brexit.
He said he looked forward to seeing greater detail on those proposals over the coming days and hoped they would be a helpful input to the negotiation process.
He noted that time was running out and the Government would engage constructively with the Barnier Task Force and the other Member States over the coming weeks.
The Taoiseach made clear that the Government was open to proposals, which met our aims of avoiding a hard border and maintaining free trade with the UK, while respecting the EU single market and customs union.
Mr Varadkar and Ms May agreed that they, and their offices, would maintain close contact in the period ahead.
Minister for Foreign Affairs Simon Coveney is calling Brexit negotiations to be intensified as "time is short".
Mr Coveney said he too welcomed the UK government was finalising specific proposals for the future EU-UK relationship.
“Ireland’s position is that we want to see the closest possible relationship between the EU and the UK,” he said.
“On the backstop for the border, our position remains clear. While our preference is still for an overall EU-UK relationship which would resolve all issues, it remains essential that a backstop is agreed which provides certainty that in any circumstances, and no matter what the outcome of the negotiations on the EU-UK future relationship, a hard border will be avoided.
Mr Coveney said talks needed to be intensified.
"Time is short. I am in regular contact with Michel Barnier and my officials are working day-in and day-out with the Task Force team he leads. Our entire focus now is on ensuring a Withdrawal Agreement, including the Irish backstop, is agreed. This will give businesses across our islands certainty in terms of planning for a status-quo transition when the UK leaves the Union in March 2019," he said.
Mr Coveney said the UK statement needed “detailed consideration” and looked forward to the additional details the White Paper would give.
Negotiations on Britain's withdrawal agreement with the EU have been deadlocked on the issue of the Border backstop, with Brussels saying it must apply only to the North and London calling for a UK-wide solution to avoid creating barriers within the UK.
In a statement issued late on Friday, the Irish Government said that EU ministers would consider the UK proposals at a meeting in Brussels on July 20th.
