The Seekers' singer died in palliative care on Friday night aged 79 after complications from a long-standing lung disease.
Victorian Minister Lily D'Ambrosio on Sunday offered her condolences to the family but said the government would not be drawn on a state funeral at this time.
"Judith Durham was a very, very private citizen and her family are very, very private too," Ms D'Ambrosio told reporters.
"Today is about remembering her and honouring her memory. She was an absolute treasure.
"Those other considerations are a matter for the coming days."
Durham made her first recording at 19, and achieved worldwide fame after joining The Seekers in 1963.
The group of four became the first Australian band to achieve major chart and sales success in the United Kingdom and United States, eventually selling 50 million records.
Durham embarked on a solo career in 1968, recording with The Seekers again in the 1990s.
Her bandmates - Keith Potger, Bruce Woodley and Athol Guy - said their lives had been changed forever by losing "our treasured lifelong friend and shining star".
"Her struggle was intense and heroic, never complaining of her destiny and fully accepting its conclusion. Her magnificent musical legacy Keith, Bruce and I are so blessed to share," they said in a statement on Saturday.
Durham's sister Beverley Sheehan spoke of the siblings' shared love of music.
"Judith's joy for life, her constant optimism, creativity and generosity of spirit were always an inspiration to me," Ms Sheehan said.
Durham's death in palliative care after a brief stay in Melbourne's Alfred Hospital was a result of complications from a long-standing chronic lung disease, Universal Music Australia and Musicoast said in a statement on Saturday.
"This is a sad day for Judith's family, her fellow Seekers, the staff of Musicoast, the music industry and fans worldwide, and all of us who have been part of Judith's life for so long," The Seekers management team member Graham Simpson said.
Tributes flowed for the beloved singer, with Prime Minister Anthony Albanese hailing Durham as "a national treasure and an Australian icon".
"Judith Durham gave voice to a new strand of our identity and helped blaze a trail for a new generation of Aussie artists," Mr Albanese said on Twitter.
"Her kindness will be missed by many, the anthems she gave to our nation will never be forgotten."
Opposition Leader Peter Dutton paid tribute to Durham as someone who "gave voice to more than one generation of Australians through words of universal appeal, carried by melodies that, once heard, became fixed in our memories".
Australia's entertainment industry also responded to the sad news, with singer Anthony Callea posting: "the skies above just gained a voice of an angel".
Actor Magda Szubanski sent her "deepest condolences".
"Her beautiful, crystalline voice was the naive but knowing siren song of my childhood," Ms Szubanski tweeted.
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