Ardern and Morrison, who is on his first visit to New Zealand as Prime Minister, had lunch together before formal talks at Government House in Auckland.
The issue of News Zealand's offer to take 150 asylum-seekers from Nauru and Manus islands was raised in their talks and their press conference afterwards.
Ardern told journalists that reports in Australian newspapers that New Zealand did not want to take single male asylum-seekers was not true.
She said New Zealand, like the UNHCR, put priority on women and children but she had not ruled out single males.
Morrison said Australia appreciated the friendliness of the offer - which was first made six years ago. But Australia would not be taking it up.
The pair deflected questions about a threat by the United States to stop information-sharing with countries using Huawei equipment and what that might mean for the Five Eyes intelligence network both countries belong to.
US Secretary of State Mike Pompeo told Fox Business Network that if a country put Huawei equipment in their information network "we won't be able to share information with them".
"We are not going to put American information at risk."
Ardern simply referred to a process being underway by the GCSB in New Zealand to assess Huawei's involvement in Spark's 5G plans.
Morrison said Australia would make its own decision in its national interest "and work with those who share an outlook on these matters".
Referring to the issue of China having been raised in their talks, he said: "We both welcome China's economic development. We think that's a good thing and we want to see that continue because it has meant a lot for our economies as well."
Ardern presented Morrison with a Warriors NFL jersey to remind him that his team, the Cronulla Sharks, had "stolen" Shaun Johnson from the Warriors.
He gave her gifts for 9-month-old daughter Neve - a koala toy, booties and a miniature Cronulla jersey.
Morrison also met Opposition leader Simon Bridges today, as is customary on such occasions.
Morrison and his wife, Jenny, were welcomed to Government House with a powhiri and honour guard before lunch.
Ardern's partner, Clarke Gayford, hosted Jenny Morrison for the afternoon.
