Morrison arrived last night and met with PNG Prime Minister Peter O'Neill ahead of a visit to the Manus Island detention centre today.
He reportedly also met with Attorney General Kerenga Kua in Port Moresby to discuss Australia's border protection policy to send asylum seekers arriving in Australia by boat to PNG.
"We are very grateful, particularly, of the role PNG has played over many years ... in assisting us in dealing with the issues of illegal arrivals to Australia and particularly the processing in offshore places," Morrison said on his arrival.
"The previous government had put some measures in place. I think we have many challenges ahead to ensure we can get those arrangements on a stable footing, an enduring footing.
"Today we can get down to some of those nuts and bolts operational issues."
A total of 780 asylum seekers are housed in Manus.
Australia had been sending asylum seekers to Manus Island since November 2012 after an agreement was struck between O'Neill and former Australian prime minister Julia Gillard.
The plan was re-installed by former prime minister Kevin Rudd and O'Neill in July this year.
Under the new agreement, all asylum seekers arriving in Australia by boat will be sent to either Nauru or PNG for processing and resettlement.
A spokesman for Papua New Guinea's Immigration Department previously warned that the arrangement PNG had with the previous Labor government stood and PNG did not want to see the Coalition taking "any shortcuts".
Kua said he was eager to discuss the issues surrounding Manus and PNG.
"Our relationship is with the Australian people, essentially," he said.
"Governments on both sides come and go but the relationship between the people is always a constant and that will drive the relationship between governments."