Police are probing whether the bat was used to assault Steenkamp, who was shot four times, including in the head, during the early hours of Thursday, or if she may have used it to defend herself, the newspaper said.
"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 fired three more shots," the police source told City Press.
Authorities have rejected suggestions that Pistorius mistook Steenkamp for an intruder and on Sunday police spokesman Neville Malila refused to comment on the reports of the cricket bat.
"I don't know where they got it from, but they didn't get it from any official source in the police," Malila told AFP.
South Africa's Olympic sprinter Oscar Pistorius (C) leaves the Boshkop police statio in Pretoria East, to be taken into police custody on charge of murdering his model girlfriend Reeva Steenkamp. (AFP Photo)
Other reports suggested there had been an altercation between the two lovers that had spilled over from Wednesday night, when neighbours called security guards complaining about a "commotion" inside the Pistorius home.
The British Sunday People said a sobbing Pistorius had phoned his friend, Justin Divaris around 4:00 am (0200 GMT) on Thursday telling him "My baba, I've killed my baba (baby in Afrikaans). God take me away."
Pistorius was 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.
His bail hearing on Tuesday will come the same day a funeral will be held for his slain girlfriend. He faces a life sentence if convicted of premeditated murder.
His father Henke Pistorius, 59, was quoted by Britain's Sunday Telegraph as saying 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."
When contacted by AFP, the father denied speaking to the Sunday Telegraph and refused to provide any further details. The newspaper's reporter Aislinn Laing confirmed she had spoken to the father on Saturday.
Echoing the family's line, Pistorius' 89-year-old grandmother also indicated the shooting had been accidental.
"I know it was a mistake anybody can make," Gertie Pistorius told AFP. "I have got my trust in my (grandson) and I have got my trust in the case, and I am sure things will go the right way" she said.
Police said there had been several previous domestic incidents at Pistorius's home.
Pistorius, who broke down sobbing in his first court appearance on Friday, has built up a powerful team of lawyers, medical specialists and public relations experts for his defence.
Stuart Higgins, a former editor of British tabloid The Sun, whose lengthy list of clients includes British Airways, Chelsea FC and Manchester United football club, will be handling public relations in the case.
One of the lawyers, 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.