He said the project would be transparent, and all the other states and territories would observe the outcome.
"[This is] so they are in a position to pick up that project on the completion of a successful pilot programme," Morrison said.
"We will work with them [SA] now to set up a time frame of how that will be put together and transparency arrangements for the other states and territories."
Morrison said it was "an important development" that could take significant pressure off of the quarantine system, if successful.
Premier Steven Marshall said the state's response to the pandemic put it in "an ideal position" to volunteer for the trial that would be under "stringent safety protocols".
He said it would likely include returning Paralympians or ADF personnel who received the vaccine in Australia.
"It will progress only under strict safety and surveillance measures and only when approved by our public health experts," Marshall said.
"While we will be the first jurisdiction to return South Australians through home quarantine, it is expected other states will then adopt the system."
Earlier on Friday, Foreign Minister Marise Payne said both SA and NSW had previously expressed interest in participating in a small-scale trial where alternative quarantine options could be explored.
"One of the things the work done so far tells us is that a vaccinated person quarantining for seven days is stronger than an unvaccinated person quarantining for 14 days," she told ABC radio.
As part of SA's public health response, it established a dedicated Covid-19 positive medi-hotel where people who contracted the virus were transferred to complete 14 days of isolation.
There are other hotels across the CBD where returned travellers whose tests were negative undergo their fortnight of quarantine.
