It comes after a deal was done ending a stalemate over Irish language laws, with the UK government saying it will legislate for language protections if the Stormont executive does not.
It is understood the Assembly nomination of Mr Givan by Mr Poots took place amid strong opposition from DUP MLAs.
DUP MPs and peers have also expressed concern.
It is understood that the vast majority of DUP MLAs – 24 to four – on Thursday morning voted against Mr Poots nominating Mr Givan as first minister at this time, during a meeting ahead of the special assembly sitting.
The nominations follow a late-night deal achieved in the early hours of Thursday after talks involving Sinn Féin, the DUP and Mr Lewis ended the stalemate, with Mr Lewis declaring Irish language legislation would be legislated for at Westminster if Stormont fails to do so by October.
But DUP MPs and peers expressed concern at the move to end a stalemate over Irish language laws.
It is understood some DUP MPs also joined the MLA meeting, but Mr Poots and Mr Givan left before the vote was taken.
Officially nominating Mr Givan, Mr Poots said the assembly needs more people like him to win for unionism.
Sinn Féin’s Conor Murphy said the new executive will face many challenges in the time ahead but that Ms O’Neill was a strong, dedicated and undaunted leader.
Accepting the nomination, Mr Givan said Mr Poots was a leader “who had a big heart for the people around him”.
He said he looked forward to his leadership.
Ms O’Neill said the new executive’s “top priority” is its response to Covid, “ensuring the continued successful rollout of our vaccination programme across the population, and rebuilding our economy to sustain jobs and people’s livelihoods”.
Ms O’Neil told MLAs the five executive parties demonstrated that “when there is unity of purpose we can tackle the biggest of challenges”.
“People want mature political leaders who will get things done.
“On the issue of rights, we have seen the DUP resist these. I don’t expect to see any change. I will work with the other parties in this assembly and the executive towards achieving a progressive social reform agenda
“Language is fundamental to nationality.”
She added that the DUP’s recent objection to providing an interpretation service for the assembly “represented bad faith”.
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