Pistorius cricket bat
The paper claims that Steenkamp's skull had been "crushed", and that forensic tests are to be carried out to determine whose blood is on the bat.
Police are said to be investigating the possibilities that Steenkamp used the bat in self-defence, or that Pistorius used it to break down a door.
The paper's police sources added that there was no sign of forced entry at the house, and that the state has a "rock solid" case against the much-decorated athlete: "City Press spoke to three sources close to the high-profile investigation, which has gripped the world’s imagination," the paper adds. "They all ruled out the possibility that Pistorius could have mistaken Steenkamp for an intruder."
The City Press report also alleges that Pistorius called his father, Henke, at 3.20am on Thursday morning, and asked him to come to his house. When Henke and other Pistorius family members arrived, the athlete was apparently carrying Steenkamp's body down the stairs. Her head and arms were reportedly "dangling", according to the paper's sources.
South Africa's Sunday Independent published an interview with an unnamed "friend" who claimed that she arrived at Pistorius's house in the early hours, and that she and Pistorius tried in vain to help Steenkamp.
Another friend of Pistorius, Justin Divaris, told the Sunday People that he too had been called by the athlete: "I said to him: 'What are you talking about? I don't understand you.' He then repeated himself: 'There has been a terrible accident, I shot Reeva.'
"Next thing his neighbour picked up the phone and told me it was true and told me to get to Oscar's home. I was in total shock. I asked the neighbour: 'Is she OK?' Did the gun go off by accident?' She replied: 'No. She's not OK. You need to get here.'"
Pistorius is due to appear in court on Tuesday when his application for bail will be heard. On the same day there is to be a memorial service for Steenkamp followed by a cremation.
Meanwhile, Pistorius’s agent Peet van Zyl has confirmed that the athlete has cancelled all his scheduled track appearances in the next few months to allow him to concentrate on defending himself.
"I have decided that following these tragic events that we have no option but to cancel all future races that Oscar Pistorius had been contracted to compete in," van Zyl, said in a statement.
The cancelled races include two appearances in Australia and one in Brazil in March, one in the United States in April and one in Britain in May.
Van Zyl, who visited Pistorius on Sunday at the Pretoria police station where he is being held, said the athlete's sponsors were sticking with him pending the outcome of his trial.
"Regarding sponsors and partners, I can confirm that at this point in time, all parties are supportive and their contractual commitments are maintained," van Zyl said.
"They have said they are happy to let the legal process take its course before making any change in their position."