DUP leader Arlene Foster met Taoiseach Leo Varadkar in Government Buildings last night where they discussed Brexit as well as efforts to restore the Northern Ireland Assembly.
The meeting lasted for more than 45 minutes. Afterwards, in a statement, they said they had agreed to stay in touch.
Earlier, Ms Foster told RTÉ News that she believes a way can be found to avoid a no-deal Brexit.
Ms Foster was in Dublin as the guest speaker at a Dublin Chamber function.
She said those who question the stance of her party should check the letter that she and then deputy first minister of the Northern Ireland Assembly Martin McGuinness sent to former British prime minister Theresa May in August 2016, in which they outlined their concerns about Brexit.
Ms Foster said she favours a solution that recognises the constitutional position of Northern Ireland, but also the fact that it has the only land border between the United Kingdom and the European Union.
"I do recognise we have a very short period of time but I very much believe that if there is a determination, a willingness and a commitment, then we can find a way through," she said.
Asked if her party had ongoing formal or informal contacts with the Irish Government, Ms Foster said: "I'm not going to give you the full breakdown of what's going on between ourselves and our neighbours, but suffice to say that we will continue to work both with our own government and with indeed anyone else who wants to find a way through all this.
"It is to our benefit in Northern Ireland and also to people in the Republic of Ireland that we find a way through."
She added: "We heard from the Taoiseach yesterday about the consequences of a no-deal situation. I say in my speech tonight that no-deal is not the optimum situation for anybody.
"We have never been no-deal Brexiteers. We have always said we wanted to find a way through, a deal that recognised what was happening on this island but also the constitutional position of Northern Ireland.
"That's all we seek and I hope that we find a willingness to seek a deal to sort all that out."
Later, she told reporters that the DUP wanted a "sensible" Brexit deal that works for people in Northern Ireland and in the Republic of Ireland.
"That's very important and one I hope that the European Union will respect," she added.
"We are in ongoing talks with our own government, our own government is negotiating with the Europeans, obviously that is the proper fashion and that should be the way it happens and we'll continue to talk.
"I think it's important there's an understanding of our position as there is an understanding of the nationalist position as well."
DUP leader Arlene Foster @DUPleader in blunt message to those who think party is softening stance on back stop. In speech to Dublin Chamber of Commerce tonight: "A deal will not be achieved that involves a backstop whether it is UK wide or NI specific." @rtenews @seanwhelanRTE— Vincent Kearney (@vincekearney) September 18, 2019
Mrs Foster added: "I've heard from Europe today ... about the majority in Northern Ireland wanting to have the backstop.
"But if you look at the Belfast [Good Friday] Agreement, it's about parallel consent, it's about the consent of nationalism and the consent of unionism.
"There are no unionist MLAs who support the backstop at present so therefore there is a need to find a way forward which everybody can buy into."
Additional reporting PA