The prime minister met leaders of Northern Ireland's other political parties on Thursday, some of whom had voiced concerns that a tie-up could destabilise local politics and undermine the British government's neutrality in overseeing separate talks to form a new power-sharing government in Northern Ireland.
"We've had some very good discussions today and those discussions are continuing this afternoon, " she said.
The European Parliament, which named former Belgian prime minister Verhofstadt as its Brexit pointman a year ago, will have the final say on any deal on Britain's exit from the EU.
Barnier has said he wants to wrap up a Brexit deal by October 2018 so it has time to get through national parliaments and the European Parliament in time for Britain's departure from the bloc at the end of March 2019. The Conservatives are considering an arrangement in which the Northern Ireland party backs Ms.
The DUP has come to the fore since the General Election after the Prime Minister had sought to strengthen her hand for Brexit negotiations by calling a snap election in April, but ended up losing her party's majority in a shock result. They have struggled for years with Irish nationalists, who want the British province to join a united Ireland.
The EU will keep the door open for Britain to return, but only on worse terms than it now has, European Parliament Brexit negotiator Guy Verhofstadt said Wednesday.
With former prime ministers stepping all over the turf, and mainstream media barely concealing its ambition to discredit the new DUP-supported, Tory-led government from the get-go, there is one way that Theresa May could clear the air, give herself some much needed legitimacy, and silence the critics. She also signalled she's willing to rethink her approach to Brexit.
That border will become an external European Union border after Brexit and there have been growing fears that any border controls would have a serious economic impact on both Northern Ireland and the Republic.
Despite the uncertainty over her ability to govern, May had confirmed that Brexit negotiations - expected to be the most complex worldwide talks Britain has held for decades - would begin as planned next week.
Talks between the DUP and the Tories are set to continue again tomorrow.