Medecins Sans Frontieres (Doctors Without Borders) confirmed on Wednesday its international staff have left the Pacific island, five days after the organisation was told its services were being terminated.

The non-government organisation says it is deeply concerned for the health and well-being of its patients on Nauru, describing the mental health of asylum seekers and refugees as "beyond desperate".

Home Affairs Minister Peter Dutton said MSF, which began its work in November 2017, was not involved in providing medical services to immigration detainees on Nauru.

Mr Dutton said dozens of other health professionals were contracted to perform that task.

Instead, he suggested the organisation provided care to Nauruans upset at the level of service provided to refugees and asylum seekers.

"Nauru, at the end of the day, is a sovereign nation as we are, and they will make decisions and be accountable for those decisions but they are an important partner for us," Mr Dutton told the National Press Club in Canberra.

But MSF later confirmed the memorandum of understanding with the Nauruan government was clear that refugees and asylum seekers would be provided services.

"Beneficiaries of the project: People suffering from various mental health issues, from moderate to severe, members of the various communities living in the Republic of Nauru, including Nauruan residents, expatriates, asylum seekers and refugees with no discrimination," the agreement stated.

MSF found during its exploratory mission ahead of the memorandum signing that the mental health needs on Nauru were "significant, and there was insufficient capacity to address them on the island".

They identified cases of schizophrenia, family violence and concerning levels of depression, especially among children.

Mr Dutton said the day-to-day management of the portfolio area had been passed on to Immigration Minister David Coleman.

Australian Associated Press