He would not say which country was behind the attacks.
Asked if it was China, Mr Morrison said: "The Australian government is not making any public attribution about these matters.
"We are very confident that this is the actions of a state-based actor.
"We have not gone any further than that. I can't control what speculation others might engage in."
Mr Morrison said the investigations conducted so far have not revealed any large-scale personal data breaches.
Australia's security agencies are working closely with allies and partners to manage it.
"I spoke to (British Prime Minister) Boris Johnson last night about a range of matters, including this one and there are a number of engagements with our allies overnight," Mr Morrison said.
The ABC said government sources had confirmed the attacker was China, while Australian Strategic Policy Institute executive director Peter Jennings told The Australian it was "very clear" that the Asian giant was behind the cyber attack.
If true, it would add further friction between the two countries, which have fallen out over the origin of COVID-19, trade and most recently the sentencing to death of an Australian drug smuggler.
The prime minister also spoke to Opposition Leader Anthony Albanese on Thursday night on the issue, as well as state and territory leaders.
"Cyber attacks are a real issue," Mr Albanese told reporters in the NSW town of Thredbo on Friday.
"What the evidence is, is that these attacks are expected to be more often."
Last month a joint statement by the Department of Foreign Affairs and Trade and the Australian Cyber Security Centre said there had been "unacceptable malicious" cyber activity.
"Of particular concern are reports that malicious cyber actors are seeking to damage or impair the operation of hospitals, medical services and facilities, and crisis response organisations outside of Australia," it said.
Last year, Australia's intelligence agency determined China was responsible for a cyber-attack on federal parliament and the three largest political parties before the May election.
Mr Morrison's vagueness about the current threat and its source was deliberate, according to Ben Scott, a former Australian intelligence official now with the Lowy Institute, a think tank.
"Public attribution – and the threat of doing so -– is seen as one way of warning and deterring an opponent," he told AFP.
"But early attribution can also be provocative," he added, saying China was "almost certainly" behind the attack.
"Australian agencies may hope that the PM's statement will deter the attackers from moving on to extract large volumes of information or engaging in any sabotage."