Representatives from the agency have travelled to the island to see conditions first hand.
They have handed a report detailing their concerns to the Federal Government.
The Red Cross already independently monitors Australia's domestic detention network and is considering providing a similar role in offshore processing centres.
Spokesman Michael Raper said the Red Cross was still considering whether to provide a regular presence on Nauru.
He has acknowledged external monitoring is needed.
"It's very important that there be that independent, neutral and impartial presence in immigration detention centres so we can intervene on individual cases immediately on the spot," he said.
"Also so we can take up systemic problems with DIAC (the Department of Immigration and Citizenship) and the Government to try and have the circumstances under which these people are detained improved."
Mr Raper said there were a number of factors to consider in taking on the role.
"It's not just a simple matter because we don't have a Red Cross in Nauru and we don't have relations with the Nauru government," he said.
"All of this will have to be established. We'll have to weigh it all up.
"There's enormous cost. We've got to see what the conditions are and what the likelihood of our presence being valuable would be so yes, we will certainly weigh it all up."
This week, Amnesty International described the conditions at the Nauru detention centre as appalling.
One of the inspectors, Dr Graham Thom, said overcrowding and a sense of hopelessness were contributing to physical and mental problems.
Meanwhile, the Salvation Army has given a relatively positive report of conditions at Australia's Manus Island detention centre in Papua New Guinea.
The group is responsible for providing welfare at the centre, as well as at Nauru.
The Salvation Army's director of offshore missions, Paul Moulds, has told the ABC radio's Saturday AM program, the situation on Manus Island is significantly better.
"I think the children are doing very well. We've both educational activities for the adults and for the children, there are some good rooms to do those things in and I think things are much better at this point in time," he said.