Perry Kouroumblis was arrested at Rome's Leonardo Da Vinci Airport in September over the slaying of Suzanne Armstrong, 28, and Susan Bartlett, 27, in January 1977.
The friends were found dead with multiple stab wounds in their home on Easey Street in Collingwood.
Ms Bartlett's 16-month-old son Gregory was found unharmed in his cot.
The slayings were dubbed the "Easey Street murders" and became one of Australia's long-running cold cases.
Kouroumblis' extradition from Rome by Victorian police will require approval by a judge. (Alyssa Braithwaite/AAP PHOTOS)
AAP has been told the Italian government has approved the Australian government's request for Kouroumblis to be extradited.
Kouroumblis told Italian authorities he was "happy" to be extradited but a judge will have final sign off before Victorian police travel to Rome to escort him back to Australia.
Charges have not been laid and he maintains his innocence.
Kouroumblis' lawyer, Serena Tucci, said her client was prepared to face trial and his extradition could happen without a court appearance.
"It's very possible that the judge's signature could happen right as the guards are there at the prison, and within an hour, he's gone," she told the ABC.
Victoria Police had an INTERPOL red notice alert out for Kouroumblis on two charges of murder and one of rape.
But he was not able to be arrested in Greece due to a 20-year statute of limitation on the initiation of murder charges.
Ms Armstrong and Ms Bartlett were last seen alive on January 10, 1977, and their bodies were found three days later.
Victoria Police Chief Commissioner Shane Patton described the murders as "an absolutely gruesome, horrific, frenzied homicide".
The force offered a $1 million reward in 2017 to catch those responsible.
The women attended school together at Benalla in Victoria's north and their families said their deaths changed many lives "irrevocably".
Victoria Police declined to comment, while the federal attorney-general's office has been contacted for comment.
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