Rachel Impson, 38, appeared polite and composed during a brief appearance in Port Kembla Local Court on Monday.

The former West Wollongong woman, believed to be living transiently, appeared via video link from Wollongong Court cells and remained mostly silent during proceedings, except the occasional "thank you".

Impson has been charged with murder.

Legal Aid solicitor John Gallagher did not make an application for her release, and Magistrate Michael Stoddart formerly refused bail.

Police responded to Reddall Reserve about 9.30pm, after Impson allegedly approached nearby officers attending an unrelated matter and informed them of the incident.

Emergency services found a 39-year-old man from Lake Illawarra, suffering stab wounds, on the footpath just south of the bridge.

Despite the efforts of paramedics, the man died at the scene.

Police spent Monday combing the area around the bridge, Reddall Reserve and Picnic Island.

An area directly in front of the Picnic Island bridge was taped off and several officers disappeared into scrub on the island.

Lake Illawarra police Inspector Ron Davis said police believed the man was stabbed after crossing the footbridge into Reddall Reserve.

He said it appeared the woman had been staying on the island in a tent for some time.

"She may have been camping on the island, or living there transiently," Insp Davis said.

"We believe the man may have come over the bridge to visit her, and that's where he was stabbed."

Police allege a 30-centimetre knife was used.

Residents in nearby homes said they heard and saw nothing until police sirens arrived.

Impson will reappear in court on December 17.