Talks between the two parties continued today amid reports any announcement of a deal may now be postponed because of the tragic fire.
They are hoping to ramp up the pressure on the region's largest political party to change its stance at a time when its policies on social issues are under renewed scrutiny due to the likely parliamentary deal with the minority Tory government.
Earlier, Andrea Leadsom, leader of the lower house of parliament, said the government had agreed with Queen Elizabeth, who reads out the new government program, that the "state opening of parliament will take place on 21 June 2017".
The talks revolve around support from the DUP on a vote-by-vote basis in parliament, rather than a formal coalition government.
May's office said Northern Ireland's five main parties would take part, but a spokesman for the DUP did not immediately respond to a request to confirm their participation.
"The UK government needs to be impartial between all the competing interests in Northern Ireland". The Conservatives are considering an arrangement in which the Northern Ireland party backs May on the budget and her confidence motions.
"The risk is that Northern Ireland continues to fail to find solutions, and potentially the peace process unwinds", he told AFP.
The nationalist Sinn Fein and SDLP and the cross-community Alliance have all made clear Northern Ireland Secretary James Brokenshire can not chair the ongoing process to restore power-sharing at Stormont due to the perceived conflict of interest.
Sinn Fein officials have refused to comment on what their MPs will do.
She is now seeking an alliance with a hardline unionist party in Northern Ireland to cling on to power.
Nolan Arenado completes cycle with walk-off homer
Arenado hit the first pitch he saw from All-Star closer Mark Melancon (1-2) into the left-field seats to cap a four-game sweep. Heading into Sunday, two of the three cycles to be completed this season have been at Coors, but to a pair of road teams.
Negotiations over Britain's exit from the European Union will begin on Monday.
"There is very little time left".
"It's imperative that both governments recommit to the word, spirit and implementation of the Good Friday Agreement if there is to be any prospect of re-establishing the Executive."The power-sharing institutions collapsed because of the DUP's RHI financial scandal and the refusal of previous Tory governments and the DUP to implement previous agreements".
Speaking straight after a meeting with DUP leader Arlene Foster, the Prime Minister praised the increased diversity in the chamber, noting the record number of LGB MPs as well as the record number of BME MPs, disabled MPs and female MPs.
Brexit is another complicating factor in the mix.
Major's warning comes amid ongoing political turmoil in Northern Ireland.
Moreover, a soft Brexit stance is also likely to please the DUP, which still wants a comprehensive free trade and customs agreement with the European Union, despite its pro-Brexit stance, as well as the guarantee of a "frictionless border" with the Republic of Ireland.
London, Belfast, Dublin and Brussels all want to keep the border open, but no agreement has been reached.
"Ruth Davidson has been very clear on that, she actually went out to Northern Ireland and set that out, so they can't be in any doubt where they stand on these issues".