Jeremy Corbyn is demanding details of the deal the Tories are striking with the DUP to form a minority government, calling it a "nonsense situation".
The Labour leader also called for a date for the Queen's Speech, saying it "was the very least we need to know".
He said Labour was "united" and "ready" to form an alternative government.
DUP sources have told the BBC an announcement on a deal with the Conservatives has been delayed because of the Grenfell Tower blaze.
They said the two parties were now finalising the "terms and conditions" of an agreement after Mrs May and DUP leader Arlene Foster met on Tuesday.
But they added that the London tower block fire made an announcement on Wednesday "inappropriate" and diary commitments meant a final deal could be delayed until next week.
If a deal was to be delayed it would mean the Queen's Speech, which had originally been planned for next Monday, could be delayed by at least a week.
It could also delay the start of Brexit talks.
The Conservatives are having to rely on the support of 10 DUP MPs after they fell eight seats short of winning an overall majority at the general election.
It means that Mrs May will remain as prime minister and the DUP MPs will be central to the survival of a Conservative Party administration.
Mrs May is to meet the leaders of Northern Ireland's other main political parties on Thursday in a bid to allay fears a deal with the Democratic Unionists will undermine the peace process.
Sinn Fein, the SDLP and Alliance have all said Northern Ireland Secretary James Brokenshire cannot chair the ongoing process to restore power-sharing at Stormont due to their perception he has a conflict of interest.
Sinn Fein's Stormont leader Michelle O'Neill said: "I will be making it very clear that any deal between the Tories and the DUP cannot be allowed to undermine the Good Friday and subsequent agreements."
In addition to Sinn Fein, whose seven MPs will not take up their seats at Westminster, Mrs May will meet representatives from three Northern Irish parties who did not win any seats at the general election - the Ulster Unionists, the SDLP and Alliance Party - in separate meetings at Downing Street.
Labour has joined former Conservative Prime Minister Sir John Major in expressing concern that deal with the DUP will undermined the "rigorous impartiality" the UK government is meant to demonstrate under the terms of the 1998 Good Friday Agreement.
Mr Corbyn is also calling on the government to end the "nonsense situation" of the public not knowing what kind of deal was on the table.
He said: "We want to know what is in the deal they are offering to the DUP and we want to know when it is going to be put before Parliament. We still haven't been given a date for the Queen's Speech."
He said Parliament "could not function until it is formally opened and I think the very least we need to know is when that is going to happen".
He added that Labour was "ready as a strong, united party with a clear programme of what we want to offer to the British people to improve their lives and end this miserable period of austerity".
What's in the deal?
By BBC Northern Ireland political editor Mark Devenport
The DUP have been playing their cards close to their chest, but we know the areas they're talking about because of a DUP plan drawn up in 2015 in anticipation of a hung Parliament.
Things have moved on a bit since then with Brexit, but we do know they're looking at trying to lower the cost to the Northern Ireland executive of any move on corporation tax.
They've been of the view that leaving the EU should lessen some of the stipulations in relation to state aid that were being applied by the Treasury to Northern Ireland, and that might take down the bill that the Treasury would put on the executive if corporation tax was lowered.
We know other matters, such as cutting air passenger duty and increased infrastructure spending, have been discussed, but we haven't got any sense of the exact details of the deal.
I suspect it will be top loaded with economic rather than political matters.
Some political issues, such as altering the definition of a Troubles victim or doing away with allowances for MPs who don't take up their seats, might be included.
Other legacy matters, such as protecting former soldiers or police officers from prosecution, may feature at a later stage.
The DUP will offer support for key votes, such as backing the Queen's Speech and the Budget and opposing any votes of no confidence.
The DUP campaigned for Brexit but is also conscious that 56% of people in Northern Ireland voted to remain in the EU.
They are thought to be advocating a Brexit that does not disrupt the "frictionless border" with the Republic of Ireland.
They are also opposed to Conservative polices such as means-testing the winter fuel allowance and have campaigned for a higher National Living Wage and to restore the spare room subsidy.
Prime Minister Theresa May said on Tuesday that talks with the DUP had been productive and that Brexit negotiations would begin as planned next week.
"I think there is a unity of purpose among people in the United Kingdom," Mrs May said, following a meeting with French President Emmanuel Macron in Paris.
On Tuesday, ex-Conservative Prime Minister Sir John Major said he was "dubious" about the idea of a deal and its potential impact on the peace process.
Sir John told BBC Radio 4's World at One programme there was a danger the government would no longer be seen as an "impartial honest broker" in restoring the power-sharing arrangements and upholding Northern Ireland institutions.
Asked about Sir John's comments, Mrs May said she was "absolutely steadfast" in her support for the 1998 Good Friday agreement - which created the Northern Ireland Assembly - and efforts to revive the power-sharing executive.