The exact cause of death has yet to be made public.
Senator White was elected to the upper house at the 2022 federal election.
In February, she announced she would take leave from the Senate due to health issues.
Before entering federal politics, she worked for 10 years as a solicitor, while also serving as the assistant national secretary of the Australian Services Union from 1995 to 2020.
She also spent a 10-year period as the vice president of the ACTU.
Workplace Minister Tony Burke said Senator White had left an extraordinary legacy.
"Linda campaigned for years in the union movement for paid family and domestic violence leave," he said on X, formerly Twitter.
"As a senator, she helped make it law so no one would have to choose between safety and pay."
Greens senator David Shoebridge said the upper house had lost a "fundamentally decent, honest, empathetic and intelligent force for good".
"Right now, I'm remembering all the good she did, the principles she held and thinking of her family, friends and colleagues," he said.
Victorian state parliament upper house member Sonja Terpstra said the late senator was a dedicated and hard-working warrior.
"A strong female lead, compassionate, smart and unyielding. Will be sadly missed," she said.
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