Police say the threat was made shortly after 10.30am (AEST) on Wednesday.
"We are currently responding to an anonymous threat made against two Melbourne airports," a police spokeswoman said.
"Police are assessing the threat and working closely with the AFP (Australian Federal Police) and airport security."
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Avalon Airport chief executive Justin Giddings said a "non-specific" threat had been made against the airports about 7am (AEST) on Wednesday.
He said the scare was not a bomb threat and no flights had been cancelled or delayed.
"We've consequently increased security and we're continuing to liaise with both state and federal authorities," Mr Giddings said.
"The only difference passengers coming out to Avalon will notice is a lot more security guards."
He said he expected the heightened security to continue at the airport for at least a week.
Letters outlining the threat to the airports were sent to Melbourne newspaper the Herald Sun.
The newspaper says the threat described in the letter was explicit, even detailing the weapons that would be used in the attack.
The paper decided not to publish the contents of the letter after a request from Victoria Police.
The threats have not caused evacuations or flight cancellations or delays at either Melbourne or Avalon airports.
Police say they are still assessing the threat.
