"It takes two to tango and we're ready to dance", she said.
"Obviously also we are going to have a discussion around Brexit".
Britain is likely to enter arduous talks on its exit from the European Union without a deal to keep Prime Minister Theresa May in power as negotiations with a Northern Irish "kingmaker" party grind into a second week.
After House Speaker John Bercow was re-elected without challenge, a chastened May quipped: "At least someone got a landslide".
Jeremy Corbyn, leader of Britain's main opposition, said his Labour Party would not support May's Queen's speech in the lower house of parliament to try to force her out of power through a vote of no-confidence.
"In terms of neutrality, politically I think it is compromised by the fact that they are now in coalition with one party".
Barnier this week acknowledged "sensitivity" in London at European Union suggestions that Britain might owe it some 60 billion euros in 2019 and said sorting out the issue soon would help a trade deal: "I would like to very quickly play down this question, and find concrete, pragmatic and just solutions", he said on Monday.
The talks Tuesday with the Democratic Unionist Party follows her apology to Conservative rank-and-file lawmakers in a meeting which signaled she would be more open to consultation, particularly with business leaders demanding answers about the details on Britain's departure from the European Union.
The DUP leader said: "There's been a lot of commentary around the issues that we are talking about and it won't surprise anyone that we are talking about matters that pertain, of course, to the nation generally".
May is under pressure to take on a more cross-party approach to Brexit talks.
May's government has said its Brexit plans remain the same, and her Brexit minister David Davis will be pressing for close economic ties but a clear break with the bloc to be able to control immigration and restore sovereignty over British laws.
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"We know each other and we understand each other", she said.
The DUP leader is nearly certain to ask for greater investment in Northern Ireland as the price of a deal.
Mrs May's authority has been severely diminished after a disastrous general election which saw her lose her Commons majority and a deal with the DUP looks vital for the continuation of Tory rule.
Even the idea of an alliance is complicated, however.
However, a deal with the DUP also risks destabilising Northern Ireland by increasing the influence of pro-British unionists.
Gerry Adams led a delegation to Government Buildings including Ms O'Neill, Sinn Fein deputy leader Mary Lou McDonald and newly-elected Fermanagh South Tyrone MP Michelle Gildernew.
The stakes for May are high.
The proposed deal would see the DUP back the Conservatives in votes on the Budget and confidence motions.
As May attempts to cobble together a majority, the EU's Barnier said he will hold talks with British envoy Olly Robbins on Tuesday to organize the negotiations.
"My preoccupation is that time is passing, it is passing quicker than anyone believes because the subjects we have to deal with are extraordinarily complex", he added.