"I think it is fair to say he was inspired. He was radicalized," the acting deputy commissioner of the Australian Federal Police, Ian McCartney, told reports in the city.
"We're not saying there was direct contact. We're saying it was more from an inspiration perspective," McCartney said.
"The circumstances of how he and when he moved from having these radicalized views to carrying out this attack yesterday will be a key focus of the investigation," McCartney added.
IS, which often claims responsibility for such attacks, has also done so in this case, though it presented no evidence that the man had direct affiliations with the group.
Read more: Yazidis fear 'IS' radicals in Greek refugee camp
Known to authoritiesPolice said the man, identified as Hassan Khalif Shire Ali, 30, had had his Australian passport canceled in 2015 after an intelligence report suggested that he planned to travel to Syria. They said that an assessment had indicated that he posed no threat to national security despite holding radical views.
Shire Ali, who was shot by police and later died in hospital, was known to the Australian intelligence agency for three years. His brother is set to go on trial next year on separate terror-related charges connected with alleged plans to kill people in a New Year's Eve crowd with a firearm
Before the stabbing attacks, Shire Ali, who came to Australia with his family in the 1980s, had driven a 4x4 laden with gas canisters into the city center and set it alight with the apparent intent of causing a deadly explosion. The fire was quickly put out before any blast occurred.
Police raidsArmed officers carried out raids on two houses in the west and northeast of the city that are linked to his family and associates, but police believe there is no ongoing threat to the public.
Crowds gathered on Saturday in front of the iconic "Pellegrini" cafe to mourn and honor the man killed in the attack, Sisto Malaspina, 74, who was a co-owner of the cafe. Among those paying tribute was actor Russell Crowe, a frequent visitor to the cafe, who wrote on Twitter, in Malaspina's native Italian, "Il mio cuore si spezza" ("My heart is broken").
Two other people wounded in the attack are still being treated in hospital, but are expected to recover.
Melbourne is Australia's second-largest city and known for its thriving restaurant and cafe scene.
tj/rc (Reuters, AFP)
Each evening at 1830 UTC, DW's editors send out a selection of the day's hard news and quality feature journalism. You can sign up to receive it directly here.