A special meeting of the Cabinet will take place tomorrow morning when the Taoiseach and the Tánaiste will provide an update on the state of the Brexit negotiations.
Contacts between Dublin and London are continuing tonight to see if an agreed position can be reached on issues relating to the border on the island of Ireland.
It is understood that the Irish Government is seeking assurances on a transition period; the retention of a common travel area; the protection of the Good Friday Agreement, and a commitment to avoid a hard border.
A senior EU source told RTÉ News that there is still no agreement on the border issue amid continuing negotiations between British and EU officials in Brussels.
However, the source said there had been "progress".
The British government faces a deadline of tomorrow to put forward proposals to reach agreement on the border between Ireland and the UK, that will allow Brexit talks to move to the next stage in negotiations.
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"There's no agreement yet, but there’s progress," the source told RTÉ News.
"There is strong desire for a deal on both sides - the wording has to be right to satisfy the political commitments of the Irish Government [on how to avoid a hard border]."
It is understood that intense negotiations have continued over the weekend, involving British, Irish and EU officials.
Teams have been working on a joint draft that was produced after a particularly intense session of talks on Thursday night.
"Thursday was a big day. There was very intense negotiation on Thursday," said the source. "They’re still going on. But the intensity on Thursday evening produced some kind of draft document, a joint text between the British and the EU."
It is not clear yet what the sticking points are, and sources suggest the Irish border issue may not be concluded in time for British Prime Minister Theresa May’s lunch meeting with European Commission President Jean-Claude Juncker tomorrow.
"It’s not there yet," said the source. "Words mean a lot. They’re trying to put language on the political commitments from the Irish side."
It is understood the special Cabinet meeting tomorrow has been called to decide if the Government can accept that whatever final text emerges overnight - if at all - qualifies as "sufficient progress" on the border issue.
That is the criteria by which Britain can move on to Phase II of the Brexit talks.
"We hope it will be done Monday or Tuesday," the source told RTÉ News.
If there is agreement by both the British and Irish cabinets, a special meeting of EU ambassadors would be called for Tuesday, since it will be the 27 EU member states which make the final call on whether "sufficient progress" has been met on all three key issues - the financial settlement, EU citizens' rights, and the Irish border.
The EU Commission will also take a formal position on Wednesday.
If Dublin signs off on the joint EU-British text, it will mean that the EU summit this month will most likely endorse the Commission’s recommendation and decide that the talks can move to Phase II.
That will require the negotiating guidelines, first adopted by the EU27 on 29 April, to be updated by the heads of government.
If Dublin is not happy with the text then there may be a few days for further work before the Commission gives its green light on "sufficient progress".
However, if the sticking points remain insurmountable, then the issue may remain open until the summit itself, in Brussels on 14 and 15 December.
'Unclear' if EU, UK can agree wording on border issue by tomorrow
Tánaiste and Minister for Foreign Affairs Simon Coveney has said that it is not yet clear if Britain and the EU will be able to agree on written assurances to avoid a hard border in time for tomorrow's deadline.
However, he said he was hopeful that a meeting between Mrs May and Mr Juncker tomorrow would create momentum towards a deal in time for the summit.
Speaking on RTÉ's This Week programme, Mr Coveney said: "The hope is that those meetings will result in a momentum that can be carried in to the leaders' summit the week after ... and can allow this Brexit negotiation process to open up to Phase II of discussions."
Asked if he expected an agreed text of written British assurances on the issue by tomorrow morning, he said: "Let's not run before we can walk here. Obviously, we would like that to be the case."
Earlier, he said the Government was not being unreasonable on the border issue in the Brexit negotiations.
Speaking in an interview with the BBC, Mr Coveney said the Government was simply asking questions that needed more credible answers before Brexit negotiations could move on.
He said he would like to see a solution that would involve all of the UK acting as one.
However, he said if that were not possible, both governments needed to recognise that Northern Ireland had unique challenges.
It comes as Mrs May, faces fresh calls from hardline Leave campaigners to walk away from the negotiating table if EU leaders refuse to sanction the start of Phase II at the summit.
Read: Coming days will stretch politics of Brexit to limit
A series of prominent figures in the British Conservative party have signed a letter urging Mrs May to refuse to settle the UK's "divorce bill" with the EU unless Brussels agrees to a series of new demands.
These include settling the terms of a free trade agreement "in principle" by the end of March 2018, and an end to the jurisdiction of the European Court of Justice.
There is also a call for an end to the freedom of movement to the UK for EU nationals when the UK leaves a year later on 30 March 2019.
"If the EU refuses to agree to these terms by the end of the December council, the UK - having exhausted every avenue - should suspend its participation in the negotiations and inform the EU that, unless they are prepared to talk to us seriously about a future free trade arrangement, we will revert to World Trade Organisation terms from March 30 2019," the letter states.
The letter was organised by the Leave Means Leave group.
Last week, European Council President Donald Tusk said that if the UK offer on the border on the island of Ireland is unacceptable for Ireland, then it will also be unacceptable for the EU.
