Though on the surface, Thursday's meeting with Northern Irish parties is aimed at breaking the logjam in forming a new cross-party regional government in the province, May needs broader acceptance of a Conservative-DUP deal.
But the Government will press ahead with the first round of Brexit talks two days before the official opening of Parliament.
Brexit Secretary David Davis and the European Commission's chief negotiator Michel Barnier agreed during discussions in Brussels on Thursday to start formal talks over the UK's departure on 19 June.
Mr Brokenshire insisted the Government would honour its commitments in the Good Friday Agreement and warned that time was running out if power-sharing was to be restored and a return to direct rule from Westminster avoided.
He said he emphasised to Mrs May that his party stood ready to form a new executive "without any preconditions or red lines".
Major said "a fundamental part of that peace process is that the United Kingdom government needs to be impartial between all the competing interests in Northern Ireland".
Gerry Adams said Sinn Féin had told Mrs May "very directly" she was in breach of the Good Friday Agreement.
May's Conservative Party, which failed to win a majority in an election last week, has been holding talks with the DUP on securing the support of its 10 members of parliament to pass legislation. It is the only way to unite the country and strengthen our bargaining power with the EU.
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A Downing Street spokesperson said both leaders "would continue to engage closely on this important issue".
The Conservative source said: "We're confident of getting an agreement, we're confident that the Queen's speech will be passed".
Alliance leader Naomi Long said Mrs May had "sought to give us her reassurance on neutrality".
A Conservative Party source said it was time for the government to get on with its business. I think that it's a very damaging situation.
He described the proposal from Sinn Féin, the SDLP and Alliance as a "sideshow".
Mrs May said: "Today I met Northern Ireland's five main parties with a clear focus on restoring devolved government to Northern Ireland as soon as possible".
The political fallout of the fire, which began late at night and trapped dozens of people in their beds, is also hurting May's Conservatives, who run the local council and have been blamed by the opposition for spending cuts that have starved budgets for building maintenance and safety checks.
"The Prime Minister and I will meet the five main Northern Ireland political parties in Downing Street tomorrow to build on the political discussions that restarted on Monday in Belfast", he said.
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