Stephen Bailey, 34, was taken to the Royal Melbourne Hospital on Wednesday night after he was arrested on Smithfield Road in Kensington.

Channel 9 reported that he was seen running along the road in his underwear before the arrest.

Mr Bailey has remained under police guard since and, as of early Thursday evening, had not been formally interviewed.

Mystery still shrouds the whereabouts of his mother.

Homicide detectives were called in to investigate after signs of a struggle were found inside her Mont Albert townhouse in Melbourne's east on Tuesday night.

It is understood blood was found inside the Strabane Avenue home that she shares with her son.

Ms Bailey, 59, was reported missing from her home on Tuesday night. She was last seen in "good spirits" about 9pm on Sunday.

It is understood her siblings and daughters had to force their way into the townhouse on Tuesday after being unable to contact her.

Mr Bailey was also reported missing on Tuesday night.

He was last seen leaving the Strabane Avenue home on a bicycle shortly before midday that day and was later captured on CCTV footage near Flinders Street Station in Melbourne's CBD.

He was captured again by CCTV cameras early on Wednesday morning near the intersection of Elizabeth and Bourke streets.

While Mr Bailey walked the city streets, police were setting up a crime scene at the home he shared with his mother. They seized two wheelie bins from the nature strip, as well as a mobile phone and other items.

Neighbours in the quiet, affluent suburb said Ms Bailey kept to herself. Most heard no signs of a struggle.

Homicide squad chief Detective Inspector Mick Hughes would not elaborate on what was found inside the townhouse, but said the scene was concerning.

"Let's just say we're concerned enough that we're actively involved in the investigation," he said.

Ms Bailey's daughters, Belinda and Shannon, made an impassioned plea for the safe return of their mother and brother on Wednesday afternoon.

In a message to her brother Belinda said: "Please come forward, we've been trying to reach you. In whatever way you can, please come forward. And we love you, and just want to hear from you."

To her mother she said: "Mum, I love you, I miss you, and I'm worried about you. If there's anything you can do to get in touch, please do. I love you. We love you."

Anyone with information is asked to contact Crime Stoppers on 1800 333 000.

- with Carolyn Webb