A man is on the run after being chased from a Melbourne house by a couple who had returned home to find him sleeping in their bed.

The man, aged in his 20s, has been dubbed a "modern-day Goldilocks" by Victoria Police after he broke into the Donvale home on December 29 and dragged the couple's mattress into the living room before falling asleep.

Investigators released footage of the alleged intruder, who can be seen in the video curled up in the white sheets, in the hopes that someone will recognise him.

Police said when the family woke the man, believed to be in his 20s, he allegedly pulled out a knife and threatened them before fleeing on foot towards Aquarius Court.

"It's very bizarre behaviour," Senior Constable Brendan Keane told reporters.

Police allege the suspect was armed with a knife. (Supplied)

Police said the suspect fled the home after being discovered by the owners. (Supplied)

"It doesn't appear he has actually stolen anything or (that) there was any motive to steal anything."

The suspect is described as having a Middle Eastern appearance, aged in his early 20s, unshaven with thin dark eyebrows and a script tattoo on the left side of his chest.

He was wearing a black New York Yankees cap and black shorts.