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South Africa's City Press newspaper says the bat is central to the police case against Pistorious, who was charged with murder after Reeva Steenkamp was shot four times at his home.
The newspaper says Steenkamp's skull was "crushed" and tests are expected to reveal whose blood is on the bat. The newspaper also said police requested that Pistorius, a double amputee known as the Blade Runner, undergo drug tests.
Pistorius's family have denied the murder charge. Initial reports said that he may have mistaken Steenkamp for an intruder have been dismissed by police.
In another development, the airing of a Caribbean reality TV show featuring the Steenkamp after she was shot dead has upset some South Africans.
In particular, women's rights activists criticised an edited clip at the start of Saturday night's 'Tropika Island of Treasure' in which law graduate and model Reeva Steenkamp talks about her "exit".
"I think that the way you go out, not just your journey in life but the way that you go out and you make your exit is so important," she says, leaning against a palm tree in a pre-recorded interview on the show's set in Jamaica.
At the end of the tribute, presumably recorded when she was voted off the show, she blows kisses to the camera and says: "I'm going to miss you all so much. I love you very, very much."
Rachel Jewkes, a gender and health researcher at the South African Medical Research Council (MRC), said the clips were particularly insensitive in a country where a woman is estimated to be killed by her partner every eight hours.
"There was a big question about whether it should have been shown at all, or whether they were trying to get audience ratings off the fact she had died," Jewkes said.
"These sort of quotes don't make you feel any better about the suggestion they are exploiting her death."
Show producer Samantha Moon said the decision to air the programme on Saturday as scheduled was difficult but ultimately she wanted to share the "special memories" of Steenkamp.
"Reeva was an intelligent, beautiful and amazing woman, and we feel it would be an injustice to keep that unknown from those who did not know her personally," Moon said.
Steenkamp, who was shot in the head, hand, chest and hip, according to domestic media reports, will be buried on Tuesday (local time).
Many South Africans thought the decision not to delay the show until after the funeral was wrong.
"It was very insensitive to put it on air before she was even buried," said 30-year-old insurance consultant Montle Ndlovu. "It's such a sad story. She was young and pretty and had her whole life in front of her."
The downfall of Pistorius, the first double amputee to run in the Olympics, has sent shockwaves through South Africa, where many saw him as a rare example of a hero who transcended the racial divides that linger in Nelson Mandela's "Rainbow Nation".
But the killing of Steenkamp has once again put a harsh spotlight on South Africa's frighteningly high levels of violence against women.
The country is still reeling from the murder this month of 17-year-old Anene Booysen, who was gang-raped, mutilated and left for dead on a building site.
Although sexual crime is all too common - on average a woman is raped every four minutes - the similarities to the murder of a New Delhi woman that triggered protests in India gave birth to an "Enough is Enough" campaign to halt the violence against women endemic in South African society.
The ruling African National Congress' Women's League called for the courts to deny bail to Pistorius to show the government was serious about stopping gender-based violence.
"Pistorius must be treated like any other person accused of such crimes and no special circumstances should be considered based on his celebrity status," the League said.
Pistorius is being held in a Pretoria police station until his bail hearing resumes on Tuesday. His family said on Saturday Pistorius was numb with grief and shock, and a pastor who visited him on Sunday said he was still distraught.
"The Holy Spirit gave me an order that this is now the chance to go pray with Oscar," pastor AJ Wilson told Reuters Television outside the police station.
"The (police) colonel told him that I have come to pray with him. He just cried and we all cried together."