Sinn Fein's Stormont leader Michelle O'Neill has urged the DUP to enter a new Executive.
The Northern Ireland Assembly is due to have its first sitting on Friday following an historic election result which saw Sinn Fein become the first nationalist or republican party to top the poll.
Sinn Fein are entitled to nominate the first nationalist or republican First Minister.
Read more: DUP adopt 'wait-and-see' approach to UK action on Northern Ireland protocol
However, the DUP said it will not nominate ministers until the UK Government acts over issues with the Northern Ireland Protocol.
DUP leader Sir Jeffrey Donaldson also did not rule out refusing to nominate a speaker, which would block the Assembly from being able to sit, on Friday.
Earlier, DUP MP Sammy Wilson said his party has adopted a wait-and-see approach over what the UK Government will do about the protocol.
Foreign Secretary Liz Truss later warned the EU that if it does not show the "requisite flexibility" over the protocol the UK will have "no choice but to act" alone.
Following a call with European Commission vice-president Maros Sefcovic, a Foreign Office spokesman said Ms Truss made it clear that the UK's "overriding priority" is to protect peace and stability in Northern Ireland.
She told Mr Sefcovic the protocol has become "the greatest obstacle" to forming a new Northern Ireland Executive.
Ms O'Neill said there is no reason for a new Stormont speaker and Executive not to be formed on Friday.
"Our Assembly will sit tomorrow; all parties should turn up, all parties should nominate and we should have an Executive up and running," she told the media during a visit to the Balmoral Show.
"There is no reason for that not to happen and it is not acceptable or not good enough that the DUP won't turn up or won't nominate for the position of speaker or the title of First or deputy First Ministers.
"The public here that I've been speaking to today want an Executive, want an Assembly. They want it working for them, they want money in their pockets to deal with the cost-of-living crisis and we can do all of that whilst trying to make the smooth implementation of the protocol.
"We will not be held to ransom by the Tory government, their internal fights between their own members. We are all being held to ransom here and that is not acceptable."
She added: "I think it is incredulous that the DUP would even consider not nominating tomorrow.
"The public here just voted in an election; they voted for politicians to work together, they voted for political parties to work together to put money in their pocket to deal with the cost-of-living crisis, to start to fix our broken health service.
"We want to be there tomorrow; we will be in a position to nominate tomorrow.
"It is only the DUP who are not going to do that.
"I don't think that is acceptable when they have just been before the electorate and the people have voted to have politics working for them."
Attorney general Suella Braverman is said to have approved the scrapping of swathes of the agreement, giving Prime Minister Boris Johnson legal cover to make the move, despite warnings from US President Joe Biden and European leaders not to single-handedly meddle with the terms.
On Wednesday, Mr Johnson said the Good Friday Agreement is more important than the protocol as he dismissed suggestions of any possible escalatory response from the European Union as "crazy".
Mr Wilson said he will not be satisfied "until we actually know what the Government intends to do, when it intends to do it, and what impact it is going to have".
Asked whether it is enough for the DUP to nominate a speaker and ministers during the first sitting of the newly elected Assembly on Friday, Mr Wilson replied: "Of course it's not, not until we see what is actually proposed.
"We have had these promises before and they haven't been acted on."
Mr Wilson told the BBC: "Ministers are approaching us on a daily basis now, indicating that the Government recognises that the protocol is causing a huge problem, that the EU are imposing the protocol in a way which was never, ever intended, and the Government is getting a bit fed up with that and are now prepared to do some acting.
"Let's face it, we have the implementation of the protocol in the most petty, draconian way by the EU, who are using it simply to teach the UK a lesson."
Alliance MLA Paula Bradshaw said: "The DUP's latest reprehensible power-play, designed to cover up its own failings during the Brexit negotiations as well as for the sake of personal expediency in the case of its leader, is putting thousands of our most vulnerable people at risk.
"The health service cannot be fixed without an Executive in place, taking steps to create a budget and manage work across all departments which will restore truly universal public health provision."
DUP leader Sir Jeffrey Donaldson said his team of MLAs will attend Stormont on Friday to sign the roll.
"We are committed to making the political institutions work but we're also clear that we need to see decisive action by the Government," he said.
Asked whether his party will nominate a speaker, he said: "Our Assembly group will meet and come to final decisions in all of this tomorrow morning.
"The Assembly meets tomorrow morning and parliamentary protocol determines that the first people to know are the people who actually sit in an Assembly, therefore we will make our position clear on this.
"We recognise that these issues need to be dealt with. As a party we have been decisive, we have made our position clear and will continue to do so."
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