Assistant Commissioner Andrew Crisp said this afternoon that police "reasonably believed" the threat was a hoax.
"We get threat information quite regularly. However, when it comes to threats involving aviation we take them extremely seriously," Mr Crisp said.
"Anything in relation to aviation should be taken seriously.
He said police had boosted numbers at the airports in the wake of the scare.
Mr Crisp said the airports made the decision to continue with scheduled flights.
Letters warning of a threat to Avalon and Tullamarine airports were sent to the Herald Sun and other Melbourne media outlets this morning.
The threat was explicit and talked about the exact weapons that would be used.
The Herald Sun has chosen not to reveal the full contents of the letter after a request from Victoria Police.
Police have visited the offices of the Herald Sun and took both the letter and the envelope it was sent in for fingerprinting.
A security expert says the threat highlights the vulnerabilities of Australian airports.
Homeland Security Asia/Pacific director Roger Henning said the incident shows Australian airports are vulnerable to attack because of lax security “airside".
Mr Henning said passengers and staff underwent security checks at the gates, but the tarmac was being breached.
He said the risk to airports was real with 60 known people living in Australia who had trained with extremist groups overseas.
And others who were disgruntled with authorities or unstable also posed a threat, he said.
“These things cannot be anticipated,’’ he said.
“There is a desperate need to improve the security culture.’’
Avalon CEO Justin Giddings said police advised the airport of a "non-specific’’ threat against it about 9am.
He said he didn’t believe the scare was a bomb threat.
"It’s extremely rare. It’s business as usual,’’ Mr Giddings said.
"We have increased security. There have been no cancellations, no delays, terminal evacuations or road closures.’’
A Channel 9 helicopter pilot hovering over the scene at Avalon said the incident did not appear to have affected airport operations, despite at least six police cars at the scene.
“It doesn’t appear the roads are blocked, but there is a heavy police presence,” the Channel Nine pilot Ross Barker told his network.
Jetstar and Qantas maintenance facilities are based at the airport.
A Qantas spokesman said no Qantas or Jetstar flights had been cancelled.
He said he was not aware of any passengers cancelling flight tickets in the wake of the scare.
"It's business as usual," he said.
Threats to the safety of aviation is a specific offence which attracts up to five years' jail.