Minister for Foreign Affairs Simon Coveney has urged his British counterpart to bring forward any viable alternatives to the backstop as soon as possible, his spokesman said following a 45-minute meeting between the pair in Helsinki.
Mr Coveney reiterated in the meeting with Dominic Raab that the Withdrawal Agreement, which includes the backstop, was not up for renegotiation, the spokesman said.
"The Tánaiste urged the British government to bring any viable alternatives to the backstop - that achieve the same goal of no hard border or related infrastructure or checks and protect the all-island economy and Ireland's place in the EU single market - forward as quickly as possible," the spokesman said.
Earlier, Mr Coveney said British Prime Minister Boris Johnson is being "totally unreasonable" over Brexit and the UK has not made credible proposals to replace the backstop.
But the British government insisted it had offered ways of solving the key sticking point.
More than three years after the country voted in a referendum to leave the European Union, Britain is heading towards a showdown with the bloc over its plans to leave on 31 October.
Speaking on Newstalk, Mr Coveney said Mr Johnson's team had not offered any concrete alternatives to the backstop.
"Boris Johnson is outlining a very clear and firm position but it is a totally unreasonable position that the EU cannot facilitate and he must know that," Mr Coveney said.
In separate remarks to reporters on arriving in Helsinki for talks with his EU counterparts, he said: "We all want to get a deal but, at the moment, nothing credible has come from the UK government in terms of alternatives to the backstop."
"If there are alternatives to the backstop that do the same job, well then let's hear them. And if we can work out a deal on that basis, so be it. But it's got to be credible."
Britain has said technological alternatives to border checks should be possible by the time the backstop might be needed, but has given no details, prompting scepticism from the EU.
Mr Coveney described such alternatives - sometimes referred to by the British government - as "vague", and said every time Dublin asked for more detail "the answer isn't convincing. In fact sometimes you don't get an answer at all".
Asked about Mr Coveney's remarks, British Transport Secretary Grant Shapps insisted Britain had made alternative proposals to replace the backstop, and it was wrong of Ireland and other EU countries to suggest otherwise.
"It's just not true," he said.
"So it's a cover when they keep saying: 'You are not putting forward ideas'- we are putting forward ideas."
Mr Johnson insists he must have the backstop removed to convince the UK parliament to ratify the deal. The EU has said it is willing to listen to London's ideas.
He repeated his commitment to the 31 October Brexit deadline.
The referendum result must be respected.— Boris Johnson (@BorisJohnson) August 30, 2019
We will leave the EU on 31st October. #LeaveOct31
He warned any attempt by MPs next week to stop Brexit or delay it beyond 31 October would do "lasting damage" to public trust in politics.
He said that preparing for a no-deal Brexit would allow the government to clinch an agreement with Brussels.
"I'm afraid that the more our friends and partners think, at the back of their minds, that Brexit could be stopped, that the UK could be kept in by parliament, the less likely they are to give us the deal that we need," he said.
British and EU negotiators will hold twice-weekly talks next month to rework the Withdrawal Agreement, which the UK parliament has repeatedly rejected due mainly to opposition to the backstop.
On Mr Johnson's suggestion of negotiations for two days a week, Mr Coveney said: "I'm sure if he wanted five days of negotiations a week, the EU would be OK with that.
"Michel Barnier is there as the chief negotiator for that purpose, he has a team that's ready to go.
"We all want to get a deal."
Mr Coveney's Dutch, Belgian and Luxembourg colleagues expressed concern over the threat of a no-deal Brexit after Mr Johnson's government moved to suspend the UK parliament, a move that outraged opposition MPs and many others in Britain.
"I am worried because for me it is a catastrophe for each and for everyone and it is a shame for the thousands and thousands of workers and create a lot of misery inevitably," said Luxembourg's Jean Asselborn.
EU leaders are due to meet for a summit in Brussels on 17-18 October, just a fortnight before Britain's current leave date.
But Brexit watchers in Brussels already expect another emergency summit might be needed in late October to avoid the worst-case scenario.
Arriving in Helsinki, Mr Raab said that accusations that Mr Johnson's manoeuvre was a constitutional outrage were "nonsense".
Ahead of his meeting with Mr Raab, Dutch Foreign Minister StephBlok said: "I'm very anxious to learn from my colleague of what their proposals to avoid no-deal Brexit would be."
"We would very much like to avoid a no-deal Brexit, it would be very bad for the UK but also for Europe and the Netherlands.... The main issue is how to deal with Irish border and you can't solve that without details," he said.
"The Withdrawal Agreement stands as it is."
Meanwhile, a Conservative MP at the centre of legislative efforts to block a no-deal Brexit on 31 October said he still believed there was time to succeed.
Oliver Letwin said he hoped MPs from across parliament would work together next week to force Mr Johnson to delay Britain's departure from the European Union. He said he did not yet know whether there was a majority for such action however.
Asked if he thought there was enough time in parliament left to prevent a no-deal exit, Mr Letwin said: "I believe that there probably is time, yes. Whether we can get the required majorities, of course, in the Houses of Commons is altogether another matter".
Additional Reporting PA