The CEO of World Vision Claire Rogers said, "these children have been forced to see and endure things that no child should ever see".
"The clock is ticking. This harmful, secretive and dysfunctional system of indefinite detention must end," she said.
In a statement, World Vision said, "the plight of the children on Nauru has largely been hidden from ordinary Australians' eyes as they languish on an island the same size as Melbourne Airport".
Kelly Nicholls from the Refugee Council of Australia said there was a deep concern for all those detained offshore, but for children in particular.
"There have been consistent reports of children attempting suicide and self-harm," she said.
Kelly Nicholls, Refugee Council of Australia
There have been consistent reports of children attempting suicide and self-harm.
"Worryingly, in recent months, there has been an escalation of reports of children suffering 'resignation syndrome' where they aren't leaving their beds and they've stopping eating, drinking, talking - even going to the toilet."
Oxfam CEO Helen Szoke said Australia needed to ensure people seeking asylum were never again subjected to indefinite detention.
"We are a nation with a strong economy, capacity to resettle a large number of people and a proven history of managing resettlement effectively," she said.
"We can and must do more."
SBS News has contacted the Department of Home Affairs for comment on the campaign and support services offered to refugee and asylum seeker children on Nauru.
The government has previously said that the offshore policy is designed to deter people embarking on treacherous sea journeys.
Labor joins chorus
Labor has also joined the chorus, penning a letter on Monday urging Home Affairs Minister Peter Dutton to accept New Zealand's standing offer to resettle 150 refugees.
"Labor is seriously concerned by reports regarding the health and welfare of children in the Australian-funded regional processing centre on Nauru," opposition immigration spokesman Shayne Neumann said on Monday.
"If Peter Dutton is too distracted by his leadership ambitions to address his failure to manage Australian-funded regional processing centres or negotiate other third country resettlement options, it's time for Malcolm Turnbull to step in and clean up his minister's mess."
Readers seeking support and information about suicide prevention can contact Lifeline on 13 11 14, Suicide Call Back Service on 1300 659 467 and Kids Helpline on 1800 55 1800 (for young people aged 5 to 25).
Additional reporting: Nick Baker