He said his party would oppose any deal between the British government and the DUP that undermined the 1998 accord but he said Sinn Féin would support any additional funds for the north secured in return for supporting Mrs May.
She is also trying to contain outrage at home over a London tower block fire which left at least 30 people dead.
DUP leader Arlene Foster said she is ready to "tango" with Sinn Fein in order to finally put a government together in NI after five months of legislative bedlam.
Barnier held "talks about talks" with May's Brexit advisor Olly Robbins and British EU Ambassador Tim Barrow in Brussels on Monday but they failed to agree on a date for the negotiations to begin, an EU official said.
The others have ruled out letting May negotiate in person with German Chancellor Angela Merkel, French President Emmanuel Macron and others.
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.
Speaking as he arrived for a meeting of European Union finance ministers in Luxembourg, he said: "As we go into that negotiation, 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 and protecting prosperity as we enter those negotiations and take them forward".
Britain said on Friday that no deal could be struck on exiting the European Union unless the future relationship with the bloc was taken into account.
A senior Brussels official said the amount, which compares to London's annual net European Union payment of around 10 billion euros, would still be "peanuts" in terms of the overall economy and also that the final bill would be determined less by technical and legal arguments than by hard-headed political horse-trading.
But pressure was mounting for May to change course on the type of Brexit Britain should pursue.
"Like Alice in Wonderland, not all the doors are the same".
He warned that a deal between the Conservatives and DUP would be in breach.
Take a look at Ireland's first social chat show - featuring live studio guests John Connors and Mark O'Connor of new smash Irish movie, Cardboard Gangsters; Matthew McConaughey sits down with Niall Horan, there's music from The City And Us, crackin" new comedy from Shifts and Joe Gorman is taught the ways of the farm by the legendary John "The Bull' Hayes..
Brexit talks will begin Monday.
It is understood there's "broad agreement" on the principles of the Queen's Speech - now due on Wednesday - which will set out the government's plans.
May will also be reliant upon the 10 lawmakers from the euroskeptic DUP, who would help her edge past the 326 votes needed in parliament to avoid the government collapsing.
The Democratic Unionist Party wants to attach its own conditions onto Brexit as part of its deal with the Conservatives - including the insistence on a "soft border" between Northern Ireland and the Irish Republic.
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