Mr Turnbull, himself a republican, said momentum must come from the public, and that he did not want to face another "heroic defeat" in a referendum.
His comments, on Australia Day, come amid mounting pressure on the government to restart the debate.
The new Australian of the Year, ex-army chief David Morrison, has said it is time to "have the conversation".
In his acceptance speech on Monday night, Lt Gen Morrison said the time was right to "at least revisit the question so that we can stand both free and fully independent amongst the community of nations".
His speech came hours after all but one of of Australia's state and territory leaders signed a document in support of dropping the Queen as head of state.
Critics have said republicans have yet to propose a viable alternative to being a constitutional monarchy.
Australians voted against becoming a republic in a 1999 referendum.
Mr Turnbull - who was leading the republican movement at that time - said he had "no doubt in the future there will be another referendum, and the matter will be decided, as it should be, by the Australian people".
"But the timing of that referendum has to be right," he said, repeating his view that it should not take place until the end of the Queen's reign.
"I've led the 'yes' case for a republic into a heroic defeat once, I've got no desire to do so again," he told reporters at an Australia Day event in Canberra.
Lt Gen Morrison was named Australian of the Year at a ceremony in Canberra on Monday in recognition of his commitment to gender equality.
He famously ordered soldiers who could not accept women as equals to "get out" of the Australian military in a speech in 2013.
In his speech, he also praised the outgoing holder of the title, Rosie Batty, a high-profile campaigner against domestic violence.
He described her as "the most remarkable woman" who had "set a benchmark for us all".
The Australian of the Year is nominated by the public and chosen by a panel, in recognition of their contribution to Australian society.
As well as an Australian of the Year, there is also a Senior Australian of the Year (a doctor, Professor Gordian Fulde, this year), a Young Australian of the Year (entrepreneurs Nic Marchesi and Lucas Patchett) and Australia's Local Hero (Youth educator Dr Catherine Keenan).
PM Malcolm Turnbull has said there are more urgent issues facing Australia than the debate on becoming a republic.