However, this turmoil does not appear to have deterred May, who intends to lead Britain into talks with European Union figures next week, while back home she tries to arrange a minority government deal with Northern Irish party the DUP.
The announcement followed speculation that talks may have to be delayed because of the inconclusive outcome of last week's United Kingdom general election, which has already forced the State Opening of Parliament to be put back from 19 to 21 June.
Major warned that although Northern Ireland was a long way from returning to the violence that killed 3,600 people, he believed the peace process remained fragile almost two decades after a US -brokered 1998 peace agreement.
May needs the support of the DUP to govern, after a disastrous election result in which she lost her parliamentary majority.
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.
Talks between the parties resumed Monday in a bid to end the political deadlock.
Separately, the DUP is negotiating with Conservatives on a deal which would allow Theresa May form a minority government.
Labour Party leader Jeremy Corbyn also congratulated Bercow, adding his movement was ready to provide a "strong and stable" government should May fail to negotiate her "coalition of chaos" - a clear jab at the Conservative leader's campaign slogans.
The pound rose after the Bank of England came its closest in a decade to raising interest rates to counter higher prices after Britain's vote to leave the European Union - a move that could further squeeze Britons with big mortgages.
Different DUP sources say different things about the approach to Brexit.
Following talks in Dublin with the new Irish Prime Minister Leo Varadkar, she said that reaching a "sensible" Brexit had been the focus of their talks.
DUP leader Arlene Foster insisted that any deal with the Conservatives would not undermine the province's peace accords. But even if that game goes into extra time, European Union leaders are making clear to May that there can be no extension to the two-year period of negotiation before Britain must leave Europe in March 2019.
"Following discussions in Brussels today, both sides agreed that the formal negotiations under the Article 50 process can now start", the statement said.
Divisions over Europe helped sink the premierships of May's predecessors Margaret Thatcher, John Major and Cameron, and many of her lawmakers and party members favour a sharp break with the EU.
"The door of course is still open as long as Brexit negotiations have not been concluded", he said.
"I think there is a unity of goal among people in the United Kingdom", May said following a meeting with French President Emmanuel Macron on Tuesday.
Scottish First Minister Nicola Sturgeon said May's government lacked the credibility necessary for Brexit talks and should delay the negotiations.