The Taoiseach has said he is not particularly optimistic that the UK side will have written proposals ready next week to be put to the crucial EU summit in mid October.
Speaking in LA, he said there is still time remaining but it's running short for British diplomats to put forward plans that are workable and legally binding that will address the Irish border question.
Earlier, Tánaiste Simon Coveney said the UK has yet to provide the EU Brexit task force with a "serious proposal" on alternatives to the backstop, adding that "significant gaps" remain between the two sides.
Mr Coveney made the comments in Brussels today, following a meeting with the EU's chief negotiator on Brexit, Michel Barnier, and his team.
Mr Coveney said: "Until there is a serious proposal, which can be the basis of negotiations, then the gaps which are wide at the moment will remain."
He added that "time is running out" and that "the onus is on the British prime minister and his team" to put "serious proposals on the table".
Mr Coveney declared that Mr Barnier and his team are "available 24/7 to negotiate to try and get a deal done".
Asked about the possibility of the UK being granted an extension to the 31 October deadline, he said the focus now is on "trying to get a deal, quite frankly".
However, he added: "From an Irish perspective, we of course think that an extension is preferable to a no-deal. But I think there would have to be good reason behind that to ask for an extension."
Yesterday, Mr Barnier gave a downbeat assessment, telling EU ambassadors that Brussels was still waiting for workable alternative proposals from Britain on the backstop, to manage the border in Ireland after Brexit.
Meanwhile, British Prime Minister Boris Johnson has refused to stop calling a law blocking a no-deal Brexit the "surrender act".
The Conservative Party leader said it was "impoverishing political debate" if such descriptions could not be used.
The so-called Benn Law demands the PM requests an extension to the divorce negotiations if a Brexit deal has not been secured by 19 October.
Mr Johnson has regularly dubbed the legislation - drafted and voted through against the Government's will - as the "surrender act".
He has faced calls from the Opposition and from more than 100 bishops to tone down his language following death threats issued to MPs this week.
Labour MP Paula Sherriff confronted the PM in the Commons on Wednesday, telling him online abusers used identical terms to Mr Johnson, including talking about "surrender act, betrayal, traitor".
But speaking during a hospital visit in England, Mr Johnson said: "I think the threats against MPs, and particularly female MPs, are absolutely appalling and we're doing a lot of work to give MPs the security that they need.
"But then there's another question which is - can you use words like 'surrender' to describe a certain act or a certain bill?
"And quite frankly I think that you can, and if you say that you can't, then you're kind of impoverishing the language and impoverishing political debate because after all, the use of that kind of metaphor has been going on for hundreds of years."
President of the European Commission Jean-Claude Juncker has said that he and Mr Barnier are doing all they can to get a Brexit deal, and if they fail it would be Britain's responsibility.
In an interview with German newspaper Augsburger Allgemeine, Mr Juncker said he and Mr Barnier were working intensively for a deal, because it would be a catastrophe for Britain and Europe if Britain left the EU without an organised withdrawal process.
"Our chief negotiator Michel Barnier and I are doing everything possible to get an agreement."
"But if we don't succeed in the end, the responsibility would lie exclusively on the British side," Mr Juncker said.
The United Kingdom is scheduled to leave the European Union on the 31 October.
Officials in Brussels say Mr Johnson has made little progress in negotiating a new exit deal to replace the one agreed by his predecessor, Theresa May, which was rejected three times by the British parliament.
Mr Juncker added that negotiations on a future trade relationship between Britain and the EU would be greatly complicated by a disorderly Brexit.
He said: "We will want to and need to seal a free trade agreement."
"But that won't happen just like that, as some in Britain imagine. Some of the trade deals we sealed in my term of office took many years to reach."