SDLP Leader Colum Eastwood said: "The context in which the talks process is now being asked to operate in could have very serious consequences if there is any suggestion of a back room deal with the DUP".
She said: "I think there is a unity of goal among people in the United Kingdom".
Northern Ireland has been without a powersharing executive since March and without a first and deputy first minister since January.
She added that "Brexit, counter-terrorism and doing what's right for Northern Ireland in respect of economic matters" were among the issues being discussed.
But Stormont parties have a June 29 deadline to end the impasse and reach consensus on re-establishing a devolved administration in the region.
THE TORIES HAVE been told that forming a government with the DUP could jeopardise the peace process in Northern Ireland.
"The danger is that however much any government tries they will not be seen to be impartial if they are locked into a parliamentary deal at Westminster with one of the Northern Ireland parties, and you never know in what unpredictable way events will turn out, and we can not know if that impartiality is going to be crucial at some stage in the future".
Ms Foster said the meeting with Mr Varadkar was useful and pleasant.
"We remain fully committed to making the institutions work", she said.
"We believe that the withdrawal process can not be concluded without the future relationship also being taken into account".
"The talks are continuing but I think the events in London today probably will have some impact on that".
"This new arrangement is very unsettling and people are concerned and anxious about what it may mean, or what promises may be given".
"If the Government cannot even secure a deal with the DUP, how on earth can they get a deal with the EU?"
The Democratic Unionist party has said it will act as a "brace against hard austerity" in its deal with the Conservatives, as rebel Tory MPs suggested that they could torpedo unpopular policies by tacitly backing Labour amendments to the Queen's speech.
Northern Ireland's largest nationalist party Sinn Fein said it would oppose any deal that undermines a peace deal known as the Good Friday Agreement, with President Gerry Adams telling Britain: "We want to govern ourselves".
"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", Hammond said."As we enter negotiations next week, we will do so in a spirit of cooperation, taking a pragmatic approach, trying to find a solution that works both for the United Kingdom and the European Union 27", he told reporters before a meeting of the 28 EU finance ministers in Luxembourg. "What would happen then?"
While European leaders try to gauge what to expect from the Brexit talks due to begin in Brussels on Monday, May is so weakened that her own Brexit strategy is the subject of public debate in her own party, and by her potential allies. "We want to see any deal between those two parties reflect the wishes of all of the people of Northern Ireland, not just one section of the community".
In a statement the European Commission said: "The opening of negotiations at political level next week will focus on issues related to citizens' rights, the financial settlement, the Northern Irish border and other separation issues, as part of the sequenced approach to the talks".