Update 6.10pm: The Taoiseach has said that the key issue of agreeing the backstop deal is not only an issue of substance but "an issue of trust".
Leo Varadkar reiterated Ireland's position on the backstop, which he said must be legally operable and would prevent a hard border emerging between the North and the Republic.
The Taoiseach was speaking ahead of a bilateral meeting with British prime minister Theresa May in Brussels.
Both leaders spoke by telephone earlier this week, however, there were no indications they are closer to resolving the outstanding Irish border issue.
Mr Varadkar said that he welcomes fresh thinking, but added that "time is running out".
"We have always been abundantly clear about what our objectives are and they have been the same since the referendum happened, and that is that there should be a withdrawal agreement that allows us for an orderly exit of the United Kingdom from the European Union," he added.
The Taoiseach travelled to Brussels on Wednesday ahead of a working dinner with the leaders of EU 27 in which they will discuss the ongoing Brexit negotiations.
Speaking about the Irish border issue, Mr Varadkar added: "We can't have an expiry date, it can only be temporary unless and until we have an alternative agreement that also ensures us that there will be no hard border on the island of Ireland.
"That's what we agreed in writing with the European Union and the United Kingdom back in March so this is not just an issue of substance, it's also an issue of trust."
He also signalled his support for extending the UK's Brexit transition period, saying that he welcomed any proposal that would help bring about a solution.
Following reports that EU chief negotiator Michel Barnier was open to pushing the implementation period back to end in December 2021, Mr Varadkar said that negotiating a new economic and security relationship between the EU and the UK within two years would be a "real challenge".
"Certainly we're willing to listen to any proposals that might help to bring us to a solution.
"I really need to say though that any extension to the transition period couldn't be a substitute to the backstop, it would still need to have that.
"And perhaps that would give people greater confidence that it would never need to be invoked."
Update 4.20pm: The Taoiseach Leo Varadkar has said he is willing to listen to any proposals at today's EU summit that will bring about a solution to the talks.
Mr Varadkar will hold a private meeting with Theresa May before tonight's key EU summit on Brexit, where the British Prime Minister will address EU leaders.
It looks likely no deal will be reached tonight, but the Taoiseach has said he is willing to listen to all suggestions.
Mr Varadkar said: "You know fresh thinking is always welcome, you know but time is now running out and we've always been abundantly clear about what our objectives are and they have been the same since the referendum happened.
"That is that there should be a withdrawal agreement that allows for an orderly exit of the United Kingdom from the European Union, there should be a transition period so business and individuals can prepare for the changes that will take place at the end of the transition period."
Latest: Adding an extra year to Britain's transition period out of the EU could give the breathing space to work out the backstop for the Irish border.
That is according to Fianna Fáil leader Micheál Martin who says it may help to focus minds.
The EU's Chief Negotiator Michel Barnier has suggested extending the transition arrangement by a year in the hope it will make a trade agreement more likely.
That would mean the proposed backstop to avoid a hard border in the case of a bad Brexit would not be needed.
Micheál Martin says it is an idea worth considering.
"Time in terms of the British politics is an important factor, I think Britain still has a lot to work out in terms of the British political system in terms of Brexit," said Mr Martin.
Simon Coveney also reacted positively to reports that EU chief negotiator Michel Barnier was open to pushing the implementation period back a year to end in December 2021.
However, British government sources insisted Downing Street was not calling for an extension to transition arrangements as Theresa May prepares to address EU leaders as she battles to keep her faltering hopes of securing a Brexit deal alive.
With 26 EU Foreign Ministers in Luxembourg with @HMcEntee at GAC debate on #BREXIT. Absolute solidarity from all with @MichelBarnier and IRE on need for a legally operable backstop in the WA, a backstop that must apply “unless and until” another solution is found. @BrendanHowlin— Simon Coveney (@simoncoveney) October 16, 2018
However, with her party split, and some Tory MPs openly calling for her to go, Mrs May has little room to manoeuvre if she is to secure a deal which stands any chance of getting through Parliament.
Ahead of her visit to Brussels, Mrs May was able to secure the backing of her Cabinet, at least for now, amid reports that some Brexiteer ministers were prepared to quit if she gave too much ground to Brussels.
However, there was anger among Tory Brexiteers after The Daily Telegraph reported that Chancellor Philip Hammond warned the meeting that the UK could still have to pay the EU up to £36 billion of the £39 billion "divorce bill" to settle its outstanding liabilities, in the event of a no-deal break.
The Prime Minister will briefly address the leaders of the EU 27 on Wednesday evening before they discuss the state of play in the Brexit negotiations over a working dinner while she leaves.
Mrs May will also have bilateral meetings with European Commission president Jean-Claude Juncker, European Council president Donald Tusk, and Taoiseach Leo Varadkar.
The EU is considering extending the two year Brexit transition period to prevent a hard border in Ireland.
The extension of one year would provide more time to develop a temporary customs arrangement between the UK and the EU.
It comes ahead of a meeting of European leaders tonight in Brussels.
Significant issues need to be resolved before a Brexit deal can be agreed, Tánaiste Simon Coveney has said.
He told BBC Radio 4's Today programme: "A lot of work has happened over the last two weeks that have actually found ways of closing gaps and getting agreement.
Referring to reports that EU chief negotiator Michel Barnier is open to extending the UK's transition period by a year, Mr Coveney said: "We have always said that we are happy to show flexibility in terms of how we get to a destination where-by there is a backstop, or an insurance mechanism, in place to reassure people on the island of Ireland that they are not going to see the re-emergence of physical border infrastructure.
"Everybody is agreed that that's where we need to get to. And, I think, Michel Barnier has shown a willingness to think imaginatively, and to show flexibility to get there.
"There needs to be a backstop that's there unless or until something better can be negotiated or agreed.
"And what Michel Barnier has indicated very clearly is that the EU side, certainly, is willing to allow more time in the transition period to agree an alternative solution to a backstop."
Ireland welcomes Michel Barnier's approach in allowing more time in the Brexit transition period to solve the Irish border issue says Deputy PM of Ireland @simoncoveney #r4today pic.twitter.com/Z5Zk7eGinS— BBC Radio 4 Today (@BBCr4today) October 17, 2018
Mr Coveney said: "What Michel Barnier is now suggesting is, look, let's ensure that the backstop is never likely to be used by creating the space and time for the UK and the EU to be able to negotiate UK-wide customs arrangements that can ensure that there won't ever be the need for customs checks between Britain and Northern Ireland.
"Which I know is a genuine concern amongst unionists.
"In Ireland we don't want that either."
Regarding setting a date for a crunch EU Brexit summit next month, Mr Coveney said: "I think what's more likely is that dates will be suggested, but that there won't be a commitment to a new summit unless there is a signal from the negotiating teams that there is something to sign off on.
"And I think that's sensible. What we don't want to do is create drama around the build-up to a new summit date and not actually have something to sign off on.
"Both sides want to get a deal done here, and I think we need... to allow the negotiating teams to set the pace with a view to making recommendations, hopefully by mid-November, that a new summit is necessary to sign off on a final deal."
The EU is considering allowing Britain to leave the EU over a longer period in return for agreeing to prevent a hard border on the island of Ireland.
It is being reported today that the idea of extending the two-year Brexit transition by one year is being considered to provide more time to develop a temporary customs arrangement between the EU and the UK.
The EU's chief negotiator Michel Barnier outlined the idea to member states in Luxembourg last night ahead of a key European Council summit later.
The meeting - which had been billed as "the moment of truth" in the negotiations - is the occasion when the leaders of the remaining 27 member states were supposed to give the green light for a special summit in November to finalise the terms of Britain's withdrawal.
Tonight's summit had been built up as the key date when a Brexit deal would get done but it is now at risk of being a bust.
EU Council president Donald Tusk has said there needs to be creativity from Theresa May when she addresses EU leaders tonight.
"The problem is clear, it is still the Irish question," said Mr Tusk.
He has warned that without new "concrete proposals" from the British to break the logjam over the so-called Irish border "backstop", further progress may be impossible.
The deal to stop a hard border with Northern Ireland if the two sides cannot reach a trade deal is what is holding things up.
Taoiseach Leo Varadkar has conceded there is lots of work to do on the backstop.
"The gaps between positions are significant and time is running out for a deal to be in place by the time the UK leaves on March 29 next year," said Mr Varadkar.
Despite this meeting being the deadline talks may now run into November or even December.
Fianna Fáil's Micheál Martin put some of the blame for the delay at the government's feet.
"It's long past time to put aside the over-produced videos and ministerial self-promotion and to show far more urgency in Brexit preparedness," said the Fianna Fáil leader.
Meanwhile, Sinn Féin's Mary Lou McDonald has hit out at the British government.
"With time running out they continue to show scant regard for Ireland, for our rights, for our economy and for our peace agreements," said Ms McDonald.
All eyes will turn to the EU court at Brussels this evening to see if an unlikely deal can be reached.