Prime Minister Jacinda Ardern says the deputy PM will be the party's deputy leader but who holds that position is up to the caucus.
Currently it is Te Tai Tokerau MP Kelvin Davis but it is unclear if he wants to remain in the job.
His performance as Acting PM in the last term has often been the subject of Opposition attacks.
This morning Ardern said she had no intention of moving Davis from the deputy leadership but ultimately that would be a caucus decision.
"When it comes to the deputy leadership within the Labour Party, I have no intent in changing that. However, all of the role allocation I will work through over the next two weeks - those do go to a vote with our caucus."
Davis did not put his hand up for the role of deputy PM when asked this morning.
"Any decision around roles and responsibilities are yet to be made. Really, that's all I'm going to say about it."
But senior MPs threw their support behind Davis.
"I totally support Kelvin," Grant Robertson said.
"He is the deputy leader of the Labour Party and he will remain deputy leader of the Labour Party, I'm sure. The question about roles in Government is one for the Prime Minister."
Andrew Little also said Davis should become deputy PM.
"He is a very solid, very hard worker and we have a very diverse caucus and he would be an excellent deputy prime minister."
He added that it was the Prime Minister's call and he would back her call.
If anyone but Davis were appointed deputy PM, "it would not be received positively" by Maori, said Little, who is also Treaty Negotiations Minister.
Tāmaki Makaurau MP Peeni Henare said he has "nothing but respect and full confidence in Kelvin Davis".
"He was my Form 2 teacher, I think he'd make a great deputy Prime Minister."
Yesterday Ardern met with the Green Party co-leaders and this morning she said she expected to complete formal discussions next week.
The following week she would finalise ministerial positions and expected the formal formation and swearing-in of the next Government.
She again stressed the mandate given to Labour to govern alone, and wouldn't say whether she had ruled out a formal coalition with the Greens.
"We do have a very clear mandate but ... I'm interested in areas of cooperation where we can use the strengths that exist in their team."
She said a transformational government was one that brought about lasting changes rather than one that implemented radical changes.
"I am determined to bring New Zealanders with us."
She expected to discuss what ministerial roles her MPs wanted in the next two weeks. She has already said that new MPs could be given ministerial roles if they had the right expertise and experience.
She said she had spoken to NZ First Winston Peters, who installed her as Prime Minister in 2017.
"It was more of a general conversation around just really my gratitude for the fact we were able to work together over the last three years."
On the Kiwis who had flown to Australia but then quarantined after travelling to states they weren't allowed to, Ardern said it was Australia's responsibility to monitor the travellers.
But Kiwis heading across the Tasman also had a responsibility to understand where they were allowed to travel.
"And remember, no matter where you go, you will be required to quarantine when you return."