Victoria Police have issued a warrant for the arrest of Peter O'Ryan, 48, following the attack in the Gippsland town of Tyers on Saturday.
Emergency services were called to a property on Cowies Road following reports two men and a woman had been assaulted about 12.30pm.
Investigators believe O'Ryan, who is known to the victims, used an axe or a hatchet to attack the trio following an argument.
The two male victims, aged 76 and 47, were airlifted to hospital with life-threatening injuries, where they remain in critical condition.
The third victim, a 74-year-old woman, was transported to hospital with critical injuries and remains in stable condition.
Detectives are still working to establish the precise circumstances surrounding the incident.
Local police, meanwhile, are combing the area surrounding the alleged attack for O'Ryan, who is believed to have fled on foot.
Police released images of the alleged assailant on Sunday in the hope the public may recognise him and provide information about his whereabouts.
O'Ryan, who is of no fixed address, is known to frequent the area around Morwell and to take refuge in abandoned properties and outhouses.
It is possible he remains armed with an axe or hatchet, police said, adding he is not known for living in the bush.
Members of the public have been cautioned not to approach O'Ryan but instead call triple zero if they see him.
The 48-year-old is Caucasian and described as being of medium build with dark grey hair.
He was last seen wearing a black and mustard coloured jacket, visible in one of the images released by investigators, and carrying a black backpack.
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