Australian of the Year and domestic violence campaigner Rosie Batty said on Thursday the "unsafe" camps must close.
Nearly 450 staff at Mr Turnbull's former university have also expressed concern about claims of torture by government employees and contractors.
Australia intercepts all irregular boat migrants and holds them in camps in Nauru and Papua New Guinea.
Mr Turnbull has defended the rules, a central policy of former Prime Minister Tony Abbott, by saying they are tough but save lives.
In a letter to Mr Turnbull, Ms Batty said those detained in offshore centres needed care, instead of punishment.
"The centres are, by their very design, unsafe and dangerous places. These centres cannot be patched up," she said.
The academics, meanwhile, said there was "overwhelming" evidence that "repugnant acts" including torture and abuse had been carried out the camps.
This week, the Australian Broadcasting Corp aired allegations made by a Somali woman who said she was raped by two men on Nauru - and that it took police on the Pacific island four hours to respond.
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Australia's controversial asylum policy
Mr Turnbull said the rape claims raised in the report were "very alarming", but defended the government's stance on border security.
"The one thing we know is these policies, tough though they are, harsh though they are in many respects, actually do work, they save lives," Mr Turnbull said.
"We're taking a number of steps to work closely with the Nauru government to ensure the safety and the security of all the refugees living in that community."
Australia's new PM Malcolm Turnbull is facing fresh calls to shut down offshore migrant detention centres.