Climate change would be one of a number of issues that would be discussed, Mr Turnbull said.
"We will also discuss the movement of our citizens between Australia and New Zealand."
The visit comes at a time when opposition parties and National's coalition partners question how close New Zealand's relationship with Australia really is, given the fact that close to 200 Kiwis are held in Australian deportation centres.
"I am delighted to host Prime Minister Turnbull in his first overseas visit as the new Prime Minister of Australia," Mr Key said.
"I was in contact with Mr Turnbull while in New York last week and look forward to broadening our discussions and cementing the relationship in New Zealand."
Mr Turnbull and his wife Lucy will also join Mr Key and Bronagh for a private dinner on Friday evening.
"Australia is New Zealand's closest bilateral partner and we look forward to continuing the tradition of close contact between Trans-Tasman Prime Ministers," Mr Key said.
"Discussions will also include my concerns about New Zealanders being deported from Australia, advancing our single economic market and cooperation on global political and security issues."
Australian Immigration Minister Peter Dutton told reporters in Canberra today that there were 285 people in the detention centre on isolated Christmas Island, and that 40 were New Zealanders.
Australian Foreign Minister Julie Bishop met New Zealand counterpart Murray McCully and Mr Key in New York last week.
Mr Key has said that he delivered a "pretty blunt" message about New Zealand's unhappiness with Australia's policy, brought in late last year, of detaining and deporting non-Australians who have incurred a prison sentence of a year or more.
Mr Turnbull became Australia's prime minister after successfully challenging the leadership of Tony Abbott last month.
