The inquiry will examine the role of state and territory governments in managing the pandemic, as well as that of the federal coalition under former prime minister Scott Morrison.
"Clearly you need to look at the response of all governments ... the different jurisdictions," the prime minister told Sky News on Sunday.
"The pandemic exposed (the fact that) some of the issues with our federation can often be quite difficult with overlapping responsibilities."
Mr Albanese said he could not envisage a circumstance where a once-in-a-century global pandemic, and in response the largest economic stimulus Australia has ever seen, was not evaluated.
An investigation would help ensure governments learned how things could have been improved and what lessons could be learned.
"Upon coming to government we've been dealing with another wave of COVID," Mr Albanese said.
"The priority has been getting through that but my government will give consideration to an appropriate form of analysis going on."
He said a royal commission would place the entire healthcare system under the spotlight including the interaction between government departments, hospitals, GPs and pharmacies.
Meanwhile, a leading infectious diseases expert warns Australia is losing its long-term battle with the virus.
Burnet Institute director Brendan Crabb says current strategies to combat the country's caseload are not working.
"What the numbers say is that we're simply not winning," Professor Crabb told Seven's Sunrise.
"The latest wave we just had ... was the worst wave we've had this year. More hospitalisations and more deaths."
Australia reported another 10,500 cases and 36 fatalities on Sunday, with the nation on track to record its 10 millionth case within a week.
Prof Crabb says there is an attitude problem.
"We haven't quite grasped the fact that having lots of virus in our community is bad. We have to change to reduce transmission, to be intolerant of the amount of virus in our community," he said.
US researchers have been analysing people who have repeatedly contracted the virus and finding the risk of acute and chronic disease accumulates with each new infection.
LATEST 24-HOUR COVID-19 DATA:
NSW: 5031 cases, four deaths, 1962 in hospital with 50 in ICU
Victoria: 2600 cases, six deaths, 414 in hospital with 28 in ICU.
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