An Irish language act is likely to be the main sticking point in getting Sinn Féin to back any new first minister.
As the largest party in the largest assembly grouping, the DUP is entitled to nominate a new first minister once Arlene Foster stands down at the end of June.
But the assembly must agree the appointment.
In return for backing a new first minister, Sinn Féin are expected to request the the introduction of an Irish language act.
Deputy First Minister Michelle O'Neill indicated that her party will continue to push for language legislation following Mrs Foster's resignation.
An Irish language strategy, along with a commitment to an Ulster Scots language, heritage and culture strategy, was agreed as part of the New Decade, New Approach deal in January last year.
Ms O'Neill said the Executive must deliver on commitments made in the 2020 deal.
"This requires a genuine commitment from all political leaders to power-sharing and to work to deliver equality for women, for the LGBT community, for Irish language and identity and all sections of our community," she said on Wednesday.
Mrs Foster vowed in 2017 to "never" agree to an Irish language act. She told party members that "if you feed a crocodile, it will keep coming back for more".
She later softened her stance but said earlier this month that language legislation was not a priority during the pandemic.
She suggested any legislation may not be enacted in the current Assembly mandate as originally envisaged.
Irish language group Conradh na Gaeilge (CnaG) announced earlier this month it would launch a legal challenge against the Executive over its failure to implement a strategy more than four years after a landmark court ruling.
In 2017, a court ruled that the Executive had failed in its legal duty to adopt an Irish language strategy - two years after a draft strategy by then culture minister Carál Ní Chuilín was not adopted by the Executive.
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