At a joint press conference in Canberra, Mr Marape indicated some progress had already been made.
“It’s a work in progress but both governments have agreed to establish a schedule going forward and for us to find some closure on the Manus asylum seekers,” Mr Marape told reporters.
While the Australian government has ruled out taking up an offer from New Zealand to resettle up to 150 refugees from Manus or Nauru, PNG has left the door open to striking a deal directly with NZ PM Jacinda Ardern.
“Common decency will apply to those human beings that are with us in Port Moresby and Manus right now," Mr Marape said.
Mr Morrison said 260 people have already been transferred to the United States and he encouraged those that have had their asylum applications rejected to return to their home country.
"We're just going to continue to work through the issue pragmatically as we have. I think we've made extraordinary progress."
Manus Governor raises social problems
Governor of Manus Province Charlie Benjamin, who was also in Canberra for the meeting, stepped up pressure on Australia to accept New Zealand’s offer to resettle up to 150 refugees from Manus and Nauru.
“My view is for them to go to a country as soon as possible. This journey has to come to an end, I think Australia really has to step up and take this idea,” Mr Benjamin said.
But Mr Benjamin didn't see moving people from Manus Island to Nauru, where Australia also has immigration facilities, as a solution, saying that would essentially be the same as staying put.
Mr Benjamin said many refugees who had fathered children with local women on the island had now departed leaving them behind.
"That is the problem that we now have and I don't know how we can be able to solve that," he said.
PNG relationship elevated
Mr Marape is the first leader to visit Australia since the re-election of the Morrison government in May.
The two leaders have agreed to hold official talks annually in a sign of the importance of the relationship.
“This is a broad and it is a deep relationship but above all it is the relationship of family and it is the relationship of true friends,” Mr Morrison said.
Earlier, cannon fire marked Mr Marape's arrival, with more fireworks expected when the two men speak behind closed doors.
Mr Morrison invited Mr Marape as part of his "Pacific step-up" strategy, aimed at increasing Australia's economic and security engagement with its neighbours.
Australia already gives developing PNG more than half a billion dollars in aid each year.
His government has also taken issue with Australia's handling of contracts in the immigration centres.
Mr Marape will also meet with Labor leader Anthony Albanese, visit the war memorial and see Governor-General David Hurley.
His visit comes just days after Greens Senator Nick McKim was kicked off Manus Island after trying to visit an asylum seeker facility.
Senator McKim denies he had done anything wrong and had followed procedures he had undertaken on past visits.
Thousands of people also attended rallies in Australia's major cities on Saturday calling for an end to offshore detention.
Friday marked six years since the Rudd Labor government reintroduced offshore detention for asylum seekers who arrived by boat.
Mr Marape will be accompanied by his wife Rachael and several ministers on his six day tour of Australia.