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.
US shoots down Syrian Air Force fighter
When a Syrian army SU-22 jet later dropped bombs near the US backed forces, it was immediately shot by a U.S. The shootdown is the latest escalation between the USA -led coalition and pro-regime Assad forces in Syria.
GST Council sets tax rates for lotteries, hotels
As for the e-way bill, the GST Council was divided on the issue and chose to allow states to continue with their existing system. Jaitley said that no decision could be made today regarding the IGST on shipping vessels at 5 per cent. "Everything is ready.
Portugal and Mexico draw 2-2 in Confederations Cup
Mexico midfielder Carlos Vela got to the ball near the far post and sent it to the middle of the area for Chicharito's goal. Ochoa made a great diving save in the 85th to keep young Portugal forward Andre Silva from scoring with a header.
Floyd Mayweather vs Conor McGregor fight confirmed
In other words, don't expect any other MMA fighters to be competing at the event, from the sounds of it. That bout pulled in over $400 million, but Mayweather v McGregor is already estimated to eclipse that.
Amazon buying Whole Foods, making strong move into groceries
The news might come as a shock to some, but Whole Foods investor Jana Partners suggested the acquisition earlier this year. Amazon could also get a better picture of customers by marrying data from Amazon and Whole Foods' loyalty programs.
Mets fall 10 1/2 games behind Nationals after 7-2 loss
A couple of hours before the first pitch, Mets General Manager Sandy Alderson held a news conference to provide injury updates. Anthony Rizzo has two leadoff homers the past two nights for the Cubs in his first two career starts atop the batting order.
Crosby dominant in Penguins' Game 5 win over Predators
On the other hand, if the Penguins win Game 5, they'll probably get the chance to win the Cup on home ice in another Game 7. The Pittsburgh Penguins will be sticking with Matt Murray between the pipes for Game 5 of the Stanley Cup Final.
Transformers Bumblebee Spin-off Movie will be set in the 80s
Mark Wahlberg will lead the star-studded red carpet at the United Kingdom premiere of Transformers: The Last Knight tonight. The 52-year old director admitted that the movies in the franchise are "massive movies" that "take a lot out of you".