It is due to be the first order of business after the Assembly election, which saw Sinn Féin win the most seats for the first time.
The Assembly cannot function if a speaker is not elected.
Sinn Féin vice-president Michelle O’Neill accused DUP leader Sir Jeffrey Donaldson of holding society to ransom.
The election of a new Speaker needs cross-community support from both unionist and nationalist members.
A new Speaker would mean assembly business could take place for up to six months, even in the absence of a functioning executive.
The DUP has also said it will not nominate ministers to form a new executive until its concerns with the Northern Ireland Protocol are resolved.
It comes after the 5 May election cemented a majority for assembly members who accept the Northern Ireland Protocol, including the new largest party, nationalists Sinn Féin.
The protocol, part of the government’s Brexit deal with the European Union, was designed to ensure free trade could continue across the Irish border by keeping Northern Ireland aligned with the bloc’s single market for goods.
However, it has faced criticism from some unionists who argue it has undermined Northern Ireland’s position in the UK by placing additional checks on some goods crossing the Irish Sea.
Sir Jeffrey said party’s MLAs would attend Stormont on Friday to sign the membership roll, but that a decision still had to be made about election a Speaker.
Sir Jeffrey said he believed the pressure his party was bringing to bear on the issue of the protocol was beginning to have an impact.
“It’s action I want to see and until I see that action and I’m satisfied that it deals with the issues that need to be dealt with in relation to the removal of that internal border within the United Kingdom, I can’t make a decision [on returning to government] until I see what is going to happen here,” he said.
“The government has to act.”
Speaking after meeting Irish Foreign Affairs Minister Simon Coveney, Ms O’Neill said Sir Jeffrey’s comments were “not good enough” and accused the DUP of holding society to ransom.“I’ll make this call to him again today, join the rest of the parties who will be there on Friday, who will be there to elect a speaker, who will be there to nominate ministers for positions, who will be there to get down to business,” she said.
Ms O’Neill added that the protocol was here to stay and accused the British government of “pandering to the DUP”.
Naomi Long, leader of the Alliance Party, said any unilateral action by the British government would be “utterly reckless”.
“We are very clear on what they [the government] should be doing – they should be engaging very intensively with the European Union and any change they make to the protocol needs to be mutually agreed by the UK and EU,” Mrs Long said.
Before his meeting with Mr Coveney, Doug Beattie, leader of the UUP, said the Irish foreign minister knew there was a “landing zone” for sorting out unionist concerns with the protocol.
“That landing site is no checks on goods from Great Britain to Northern Ireland,” he said.
Mr Beattie said that despite these issues, it didn’t give the DUP “the excuse to stay out of government”.
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