"Our number one priority is the health, welfare and safety of all South Australians. At this stage we cannot lift that border (with Victoria) on the 20th July as we were hoping to do," the premier told reporters on Tuesday.
It comes as NSW premier Gladys Berejiklian reiterated her government is not currently contemplating a closure of the NSW-Victoria border.
But she warned NSW residents they are becoming lax on social distancing measures, with the threat of outbreak still elevated.
"Things can change very quickly in terms of the rate of community transmission.... I have noticed in and around my movements that people are starting to relax a little bit too much for my liking. Don't relax," Ms Berejiklian said on Tuesday.
"Assume everybody in and around you has the disease."
NSW Health Minister Brad Hazzard, meanwhile, said the Victorian government should look to local lockdowns in Melbourne's outer suburbs.
"We've made it clear to Victoria that we'd be much happier if the hotspot suburbs, they've obviously got some challenges, were put into lockdown effectively," Mr Hazzard told 2GB radio.
The NSW government has repeatedly warned residents of greater Melbourne to steer clear of NSW until community transmission was reduced and has announced it will turn away football fans from its southern neighbour.
Sport spectators trying to enter NSW stadiums from Wednesday are likely to be required to show their driver's licence to prove they're not from Victoria.
"They might sound like tough things to ask people to do but that's what will keep us safe in NSW and we certainly want to continue on the path we're on ... we have done extremely well," Ms Berejiklian said.
Ms Berejiklian on Tuesday also encouraged her Queensland counterpart Annastacia Palaszczuk to announce a plan to open the state's borders next month.
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