According to reports of the meeting, she accepted personal responsibility for calling the snap election and for the result, which saw her party lose its overall majority and have to rely on the support of others. It has already been indicated the ceremony could be delayed by several days.
In her opening speech in the House of Commons, May congratulated Bercow and joked: "At least someone got a landslide".
"What we're doing in relation to the talks that we're holding, the productive talks we're holding with the Democratic Unionist Party, is ensuring that it is possible to, with their support, give the stability to the United Kingdom government that I think is necessary at this time", May told a news conference in Paris following a meeting with French President Emmanuel Macron.
Foreign minister Boris Johnson, who was reported by British media to be lining up a leadership bid, insisted May should stay.
Jeremy Corbyn, leader of the opposition Labour Party, which saw its number of parliamentary seats and share of the vote increase, said there could be another election this year or early in 2018 after Thursday's vote produced no clear victor.
The political turmoil comes a week before Britain is due to start negotiating the terms of its exit from the European Union in talks of unprecedented complexity that are supposed to wrap up by the end of March 2019, when Britain actually leaves.
"There can be no backsliding from the objectives the PM set out in the [election] campaign - taking back control of our laws, our borders, our cash; but also ensuring that we have a great new partnership with the European Union that allows us to trade more freely and enthusiastically than ever", Mr Johnson wrote in the right-wing tabloid The Sun.
Over the weekend, May's top two aides stepped aside.
Turkey Earthquake Kills Woman on Greek Island of Lesbos
An natural disaster with a magnitude of 6.0 hit the western coast of Turkey on Monday, with no casualties reported so far. Geological Survey concluded that this one "occurred as a result of normal faulting in the shallow crust".
It is thought the PM will scrap plans to rethink the "triple lock", which guarantees the state pension, as well as means tests for winter fuel payments - two policies which proved unpopular during the election, and which have also been rejected by the DUP.
May also restored former Justice Secretary Michael Gove to the Cabinet in another move created to show she was willing to listen to critics. Gove, who was dismissed when May became prime minister past year, will now serve as environment secretary.
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.
Senior EU sources have told the Guardian that if London was firm on talking about a free trade deal before the issues of its divorce bill, citizens rights and the border in Ireland were resolved, then it would be met with a blunt response.
Moody's Investors Service said the inconclusive election outcome would complicate and "probably delay" negotiations with the EU.
"Hopefully we can make a bit more progress later on Tuesday", he said. "Scotland voted against Brexit. May would not have countenanced previously, and which would be positive". Labour said the government was "in chaos" and continued to be "in denial" about the message voters had sent about their opposition to an "extreme Brexit". The Institute of Directors survey said company directors see no clear way to resolve the political situation quickly.
The performance of the British economy could also influence perceptions of Brexit.
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