May's botched election gamble cost her Conservative Party its majority in the 650-seat parliament last week, weakening her hand just days before talks with the European Union on a divorce that has to be agreed before an exit due in March 2019.
BBC chief political correspondent Norman Smith says the deal "will be top loaded with economic rather than political matters", including a cut in corporation tax for Northern Ireland.
Sinn Fein's Gerry Adams turned Mrs May's own slogan against her to brand it "a coalition of chaos", adding: "Any deal which undercuts in any way the process here or the Good Friday Agreement is one which has to be opposed".
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.
The 1998 Good Friday Agreement commits the United Kingdom and Irish governments to demonstrate "rigorous impartiality" in their dealings with the different political traditions in Northern Ireland.
The first round of Brexit talks will start on Monday (19 June), officials confirmed after negotiators met in brussels on Thursday to hammer out the details of the procedure.
At least six people died and dozens were injured when a massive fire tore through a 24-storey London apartment block overnight, a fresh blow to the capital less than two weeks after the London Bridge terror attack.
Mrs O'Neill said she would discuss her concerns with new Irish PM Leo Varadkar the following day.
Mrs Foster has also rejected suggestions that the mooted agreement could undermine a return to powersharing arrangements at Stormont amid claims from political rivals that the Government's stated impartiality as a mediator would be fatally undermined.
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An initial round of talks between May and DUP leader Arlene Foster ended with no agreement on Tuesday, although both sides said they were hopeful, with talks to resume Wednesday.
Mr Brokenshire, who will also attend the Downing Street exchanges, characterised the meetings as a contribution to the three-week process to restore devolution.
The proposed deal would see the DUP back the Conservatives in votes on the Budget and on any confidence motion while other matters would be negotiated on an issue-by-issue basis.
But with so much at stake for Britain and its US$2.5 trillion economy, pressure was mounting on May heed other voices.
Major said: "We need to be prepared for the unexpected, we need to hope for the best but prepare for the worst".
They've been of the view that leaving the European Union should lessen some of the stipulations in relation to state aid that were being applied by the Treasury to Northern Ireland, and that might take down the bill that the Treasury would put on the executive if corporation tax was lowered.
Mrs May headed to Paris later Tuesday to meet with French President Emmanuel Macron, and the two leaders attended a friendly football match between France and England.
The appointment of Baker may have more to do with consolidating her position than signalling a Brexit course.
"It's a unity of objective, having voted to leave the European Union, that their Government gets on with that and makes a success of it, and we are committed to developing a deep and special partnership with the European Union".
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