The negotiations to form a new government have been going on for several weeks.
Speaking to media on Wednesday afternoon, Luxon said: "As I said yesterday we are int he final stages of our coalition conversations, and we've made tremendous progress as I've said before".
"We've agreed fully a policy programme, we've also agreed the processes and ways of working as a government, we've importantly largely worked through ministerial responsibilities and positions."
He said it was fair to say deputy prime minister is one of the sticking points, but "those are conversations I've said I'd have with the leaders".
It will take as long as it takes, he said.
"But we're so close, we've come so, so far. And we need to close it out quickly and get to work.
"Suffice to say we've got a couple of issues to close out and then we can get this thing done ... largely we've agreed our ministerial positions and appointments."
He said Nicola Willis as deputy prime minister was never a consideration.
"No, absolutely not. It had never been a consideration for Nicola, her and I both spoken well before the negotiation and of the view that frankly given our past convention that actually deputy prime ministership is a role that should be held by coalition leaders."
He continued to repeat his position that he would conduct negotiations in the room with the other party leaders.
"When you look at how MMP works around the world and you look at how long it takes to form governments in other MMP environments, I think we've made tremendous progress and the fact that we have agreed an aligned policy programme that all the party leaders can support is a fantastic achievement. The fact that we've agreed processes is really important and the fact that we've largely resolved our ministerial appointments and responsibilities is great.
"The three parties are very aligned around the goals of how to take this country forward ... we're gonna work on that together."
He said there was "incredible talent" within each of the three parties, "and as the prime minister I want to be able to use all of that talent and put the right people in the right assignments to be able to go and get the job done".
Asked who had been harder to deal with between the other two party leaders, he said he had enjoyed working with both.
"I think both gentlemen have actually done an exceptionally good job and I think all three of us have actually worked well."
He said he had been on the phone a lot since early this morning. "I've had to recharge my phone twice."
"As I've said before MMP is an environment that New Zealanders have chosen for (sic), the New Zealand voters are never wrong. Whatever result they give us on election night we are forced to make it work.
"We are determined all three of us to come together to collaborate, cooperate, work together for the interests of New Zealand. I think they've done that very well.
"Not getting into any of that, those conversations as I've kept saying - and I know it's repetitive, but they are important that I respect the confidence of that room."
He said there was "no-one holding the country to ransom".
"I'm trying to get people to understand that in an MMP process it requires some maturity and it requires serious stuff and I laid up a process that I've followed really clearly all the way through it. And we are trying to do something to make sure New Zealand gets a really strong, stable government and I'm really serious about that.
"I've laid up a process from day one which was: Policy, process, appointments. And so we've done each of those things as a component part and then moved on to the next."
He said they would continue the conversations tonight, possibly tomorrow, as long as was needed.
"But once we have agreement there we'll be able to signal that we've concluded our negotiations. They will need to be ratified by our respective party organisations which will be pretty quick, and then we'll be moving very quickly through to Wellington to announce what we've agreed.
"I think I'll close things up from up here at this stage. But anything's changing, as I said we've been agnostic to Auckland and Wellington it's just been about where our weight of teams have been, and where the party leaders have been."
Today some senior National MPs, including deputy leader Nicola Willis and Chris Bishop, returned to Wellington, with Willis telling RNZ her job in Auckland was done.
It has now been nearly six weeks since the 14 October election, and almost three weeks since the final vote total was announced. Special votes saw National and ACT lose their majority, meaning they need NZ First to form a government.
Luxon has been holding talks with ACT leader David Seymour and NZ First leader Winston Peters for several days in Auckland, and has indicated talks were nearing their end.
On Monday, Luxon said policy programmes had been agreed with ACT and NZ First and ministerial positions were being discussed.
