Mrs May needs the votes of the DUP's 10 MPs to prop up her minority administration as she hopes to steer government business, including crucial measures on Brexit, through the Commons.
With the two-year clock on Brexit ticking since March, when a letter from May formally started proceedings, Barnier dismissed the suggestion of postponing the negotiations and said such a delay would only prompt further instability.
The talks are being closely watched in European capitals as they could delay the expected start of Brexit negotiations next week, as well as change Britain's entire approach to its EU withdrawal.
One of the most pressing issues facing Mrs May is the process of leaving the European Union, with Brexit talks set to start in Brussels on Monday.
Liberal Democrat MP Alistair Carmichael said: The Conservatives can not go from weak and wobbly to business as usual in three days.
On Monday, the European Commission insisted it was "fully prepared and ready for the negotiations to start" the process.
Besides forging a deal to keep her job and preparing for Brexit talks, May is grappling with a crisis over the breakdown in power-sharing between the pro-British DUP and the Catholic nationalist Sinn Fein party in Northern Ireland's separate, devolved government.
Theresa May must immediately create a cross-party joint Cabinet committee to negotiate Brexit. The party has refused to give a time frame for reaching a deal, though May is due in Brussels for an European Union summit on June 22-23 when she will want to show she has a solid grip on power.
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