It comes as SNP leader Nicola Sturgeon voiced concerns over the prospect of a "grubby deal" between the Tories and the DUP showing "disregard" for the Northern Irish peace process. "An agreement to restore devolved power-sharing government in Stormont must be reached by the June 29 deadline", he said.
Having lost a majority in parliament, May needs the DUP's 10 lawmakers to win any kind of vote, including on the pieces of legislation needed to enact Britain's divorce from the EU.
Talks between the two are ongoing but the opening of the new Parliament is set for Wednesday.
"We want to see a Brexit that works for everybody, not just in Northern Ireland from my perspective but in the Republic of Ireland as well, so it is about a sensible Brexit", Foster told reporters in Dublin.
Ireland's new prime minister will meet his British counterpart, Theresa May, for the first time on Monday to discuss Brexit and the political deadlock in the Northern Irish regional assembly.
Ms O'Neill said: "We made very clear the the Prime Minister that any deal between herself and the DUP can not undermine the Good Friday Agreement, it can not undermine the joint nature of the office".
"We also respect the other parties' mandates, we want to get back to an executive that has all the parties around the table to collectively take decisions".More news: Grenfell Tower death toll: Number of people confirmed dead rises to 30
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Negotiations over Britain's exit from the European Union will begin on Monday. While post-Brexit Britain will remain an important export market for the EU-27, its isolation in Europe and loss of preferential access to the bloc's trading partners could have dramatic consequences.
"We made the case to her that we would oppose any deal that undermined the Good Friday Agreement", he said.
The discussions about Britain's new relationship with the European Union are expected to last for two years.
May's finance minister, Philip Hammond, said on Friday that Britain should prioritize protecting jobs and economic growth.
"My clear view, and I believe the view of the majority of people in Britain, is that we should prioritise protecting jobs, protecting economic growth, protecting prosperity as we enter those negotiations and take them forward", he said.
With #Brexit negotiations set to begin Monday, could #UK be in any worse a place? "They can't have it both ways, it has to be dealt with sensibly", she said. Her finance minister said at an European Union meeting on Friday that a Brexit deal should put jobs and prosperity first.European Union officials played down the importance of Britain's lack of a clear final plan, saying that talks on other issues can go ahead without deciding yet on a new relationship.