The Taoiseach Leo Varadkar and British Prime Minister Boris Johnson have spoken by phone to discuss Brexit.
It comes as talks between London and Brussels are breaking down, with Downing Street sources saying a deal is essentially impossible.
The German Chancellor Angela Merkel has told Mr Johnson a Brexit deal is "overwhelmingly unlikely", according to Downing Street.
Mr Johnson spoke with Ms Merkel for 30 minutes on Tuesday morning, with Mr Johnson stressing that Brexit negotiations in Brussels "are close to breaking down".
An EU-UK agreement is "essentially impossible not just now but ever" following the "clarifying" phone call, a Downing Street source added.
A spokesman for Mr Varadkar said both sides reiterated their desire to reach a Brexit deal.
He said they hope to meet in person later this week.
British media outlets are reporting Mr Johnson will again visit Dublin on Thursday or Friday.
The @Europarl_EN approach is very straight forward: we will not agree to a Brexit deal at any price. We will always fight to protect citizens’ rights and the peace process in Northern Ireland. My full statement → https://t.co/RJjqP6dGC9 pic.twitter.com/hApgopjiV5
— David Sassoli (@EP_President) October 8, 2019
Meanwhile the new president of the European Parliament has said there has been "no progress" on Brexit following talks with Mr Johnson in Downing Street.
"I came here in the confident hope of hearing proposals that could take negotiations forward," David Sassoli said in a statement.
"However, I must note that there has been no progress."
Mr Sassoli said the UK leaving with a deal was "by far the best outcome" - but the European Parliament "will not agree a deal at any price".
He added: "We have examined the UK proposals to replace the original backstop and our response is that these are a long way from something to which the parliament could agree.
In addition, they are not immediately operable."