The Conservative leader lost her parliamentary majority in last week's election and is now desperately seeking the backing of the 10 MPs from Northern Ireland's ultra-conservative Democratic Unionist Party (DUP).
EU leaders have voiced growing impatience to start Brexit negotiations, which have already been delayed by the parliamentary election -- and on which the clock is ticking.
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 of a deal.
"The talks are continuing but I think the events in London today probably will have some impact on that. I think it's unlikely there will be any announcement today," a DUP spokesman told AFP.
Media reports suggested an agreement could be delayed into next week, but the spokesman said: "I certainly have heard nothing on this side to indicate that."
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.
May's office issued a statement saying she was "deeply saddened by the tragic loss of life" but made no mention of the talks with the DUP.
On a visit to Paris on Tuesday evening where she met with French President Emmanuel Macron, May described the talks with the DUP as "productive".
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.
'Softer Brexit'
May has dismissed calls to resign following the dismal election result after calling a vote three years early in the hope of bolstering her slim majority, only to actually lose seats.
A lacklustre campaign saw her high approval rating slip away, and support for her "hard Brexit" strategy -- pulling out of the European single market and customs union -- now hangs in the balance.
Former prime minister David Cameron, who called last year's EU referendum and resigned after losing it, told a conference in Poland that "there will be pressure for a softer Brexit".
Parliament now "deserves a say", he said, adding that there was "perhaps an opportunity to consult more widely with the other parties on how best we can achieve it".
Guy Verhofstadt, the European Parliament's chief Brexit negotiator, warned Tuesday that "the current uncertainty cannot continue" and on Wednesday issued five "pressing questions" on Twitter.
Among these was whether Britain's position would "be the same as in the letter of March 29" when May triggered the two-year Brexit countdown.
The EU's chief Brexit negotiator Michel Barnier also warned Tuesday that time was passing.
"It's passing quicker than anyone believes... That's why we're ready to start very quickly. I can't negotiate with myself," he told the Financial Times newspaper.
Barnier dismissed the suggestion of postponing the negotiations and said such a delay would only prompt further instability.
At a press conference with May, Macron said the door was "always open" for Britain to remain in the EU as long as the negotiations on Brexit have not finished.
'Very unsettling'
The DUP is believed to be more favourable to a "soft Brexit" that would keep Northern Ireland's border with the Republic of Ireland free-flowing.
Under the proposed deal, the DUP would likely support May's Conservatives on big issues such as the budget, Brexit and defence legislation on a vote-by-vote-basis.
They would not form a coalition. However, the prospect of a deal has prompted warnings that it could upset Northern Ireland's fragile peace.
London's neutrality is key to the delicate balance of power in Northern Ireland, which was once plagued by violence over Britain's control of the province.
"The danger is that however much any government tries they will not be seen to be impartial if they are locked into a parliamentary deal," 1990s Conservative prime minister John Major told BBC radio.
The Irish republican Sinn Fein party -- which won seven seats in the election although their MPs traditionally do not take up their seats in protest -- is also wary of the alliance.
"This new arrangement is very unsettling and people are concerned and worried about what it may mean," Sinn Fein MP Michelle Glidernew told AFP.
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