South African police have found a bloodied cricket bat at the home of Oscar Pistorius, a local newspaper reported Sunday, following the Valentine's Day killing of his model girlfriend.

The star Olympian and Paralympian athlete was charged on Friday with the murder of 29-year-old Reeva Steenkamp who was shot dead at his luxury Pretoria home in a case that has gripped the world.

His father said in a newspaper interview published Sunday that there was "zero doubt" his son had mistaken his blonde covergirl girlfriend for an intruder, a suggestion the South African police have dismissed.

Pistorius's family has described the 26-year-old as "numb with shock as well as grief" over the death of Steenkamp, and rejected the murder allegations.

Police sources close to the investigation told South Africa's independent City Press newspaper that Steenkamp's skull had been "crushed".

"There was lots of blood on the bat," one source said.

Police are investigating whether the bat was used to assault Steenkamp, who was shot four times in the early hours of Thursday, or if she used it to defend herself.

"The suspicion is that the first shot, in the bedroom, hit her in the hip. She then ran and hid herself in the toilet... He fire three more shots," a police source told City Press.

The paper said he could have experienced "roid rage", aggressive behaviour linked to taking large doses of steroids, describing the case against Pistorius as "rock-solid".

Other reports suggested there had been a fight between the two lovers that had spilled over from Wednesday night when neighbours called security guards complaining about a "commotion" inside the Pistorius house.

Police on Sunday refused to comment on the case.

Pistorius -- a national icon who inspired people around the world when he became the first double amputee to compete against able-bodied athletes in the Olympic Games last year -- is spending the weekend in a police cell.

He is due to apply for bail at a new court hearing on Tuesday, the same day a memorial service will be held for his slain girlfriend. He faces a life sentence if convicted of premeditated murder.

"We have no doubt there is no substance to the allegation and that the state's own case, including its own forensic evidence, strongly refutes any possibility of a premeditated murder or indeed any murder at all," his uncle Arnold Pistorius said in a statement Saturday.

His father Henke Pistorius, 59, told Britain's Sunday Telegraph that the family had "zero doubt" his son shot Steenkamp on instinct after mistaking her for an intruder.

"When you are a sportsman, you act even more on instinct," he said. "It's instinct -- things happen and that's what you do."

Police have dismissed the intruder theory, noting that there had been several previous domestic incidents at Pretorius's home. City Press said she was wearing a nightie at the time of the killing and that the bed was rumpled.

Pistorius, who broke down sobbing in his first court appearance on Friday, has built up a powerful team of lawyers, pathologists and public relations experts for his defence.

One lawyer, Kenny Oldwage, acted for the driver in a 2010 accident that killed former president and anti-apartheid icon Nelson Mandela's great-grandchild Zenani. The driver was acquitted.