Mr Morrison said he will be using time ahead of Wednesday's national cabinet meeting to finalise plans with the states to respond to supply issues of rapid antigen tests (RATs).
But he said making RATs free for all households - a measure used in the UK and the US - was impractical.
He said with limited resources at hand, there would be targeted distribution of the tests in health and education settings.
Mr Morrison said on Thursday that he believed the private market should cater to the demand for the general public, including households and small businesses.
A black market is emerging in Australia for rapid antigen tests with the scarcity in supply. Source: AAP
"...for all other casual uses – you would just like to get a test or something like that – well, that is what the private market is for," he said on Thursday after the national cabinet meeting.
He said the cost of the test would be made affordable through its tax-deductible status and plans were underway for improved access for concession card holders.
"We’re just in another stage of this pandemic now, where we can’t just go round and make everything free," Mr Morrison told Channel Seven's Sunrise program on Monday.
"This is not a medicine, it is a test. And so there is a difference between those two things.
"They are available at $15 and we are working on arrangements as I flagged two weeks ago for concessional access to those who are pensioners and others.
"And we’ll be working through those issues."
Mr Morrison is himself after a positive case attended a press conference he held on Wednesday to foreshadow a snap national cabinet meeting on Thursday.
Following that meeting, a and isolation rules, which gave a bigger role to RATs over PCR tests.
The decision followed growing pressure on PCR testing clinics with long queues and blown-out turnaround times on testing results.
The high volume also resulted in over the Christmas-New Year period.
Within 24 hours of announcing the changes, the federal government had announced that the use of RATs would be further reduced so that only close contacts - not confirmed cases - would need to use the RAT on day six.
Poorest priced out as charities, community groups struggle to source supply
Charities and community groups say the short supply is affecting those who are on the lowest incomes and those who are the most vulnerable.
The Rev. Bill Crews Foundation said there has been no response to his plea for a supply of 500 tests from the Sydney Local Health District and the state government.
Instead, the charity is spending about $20,000 sourcing its own supply.
President of the Australian Council of Social Service, Peter McNamara, said free RATs must be provided to Australians who cannot afford to pay for the tests, including First Nations people, people with disability and the unemployed.
"Almost four times as many people on low incomes have died from COVID as opposed to the rich," he said.
"What we are saying is use the evidence, get access to rapid tests into the hands of the most vulnerable -community services and frontline services - who have saved thousands of lives during the pandemic."
Calls grow to make rapid tests free for everyone
Health Program Director at the Grattan Institute, Stephen Duckett, says there has been a total failure of planning by the federal government to ensure an adequate supply of RATs as part of the national COVID-19 strategy and the reopening plan.
“We should use the word farce, joke or mess to describe what the Commonwealth government has done on the (supply of) rapid antigen tests,” he said.
He said the shortage in supply in chemists and shops could have been anticipated and planned for when the Grattan Institute and others raised it as far back as June 2020.
“There has been no planning whatsoever. In England, these tests have been available for free to everybody.
“We didn't even approve them until November - and then we made this change - and all of a sudden every chemist and shop in the country was out of them."
He is urging the federal government to act immediately to make RATs free for all, saying concessional access will not solve the supply problem.
“Concessional access is not going to work because it means people have to have the indignity of proving to a chemist - or someone they don't know - that they don't have any money.
“And this is not something we want in a system of universal healthcare.”
Opposition leader Anthony Albanese urged the federal government to intervene to deal comprehensively with the affordability issue.
"There is the issue of affordability where a whole lot of people are priced out of gaining access," he told ABC Radio.
"And it is just not good enough. This government always responds too little, too late. It never prepares for what is necessary."
'Makes little sense'
Epidemiologist Marylouise McLaws from the University of NSW said the government's reluctance to make the tests free for everyone was hard to understand.
"It makes little sense," she told SBS News.
"I am very concerned that those in the community who are underemployed, unemployed and finding it very hard to put food on the table won’t be able to afford to help the community, which is a really important public health tool.
"They should be for free for the whole of the Australian community."
She said testing has been critical so far in detecting cases - and that could be missed if people struggle to access either RAT or PCR tests.
"Over the last couple of days - from the 1st of January and the 2nd of January, there were over 24,000 infected - and they were identified through testing.
"But we won’t necessarily have that information ever again (with less use of tests).
"We won’t be able to tell the World Health Organisation how many cases we have." And certainly, Australians won’t know how fast the outbreak is changing."
Federal Health Minister Greg Hunt said there are "very considerable supplies" of RATs that will be made available to concessional cardholders.
He said the state governments are providing the RATs with the cost being covered 50-50 between the federal and state governments.
"Obviously there’s high global demand, but just quietly, the Australian government has gone into the market and secured that support for those with health conditions and those that are concessional cardholders," he said.
