Police confirmed that a friend of Mr Pistorius’s was killed at his home in the early hours of Thursday morning.
They said a 26-year-old man was due in court, but this has been delayed until Friday. Police do not name suspects until they appear in court.
Mr Pistorius, 26, was the first double amputee to run in the Olympics.
The precise circumstances surrounding the incident are unclear. Some reports said that Mr Pistorius may have mistaken his girlfriend for an intruder.
But police said they were “surprised” to hear reports that the deceased had been mistaken for a burglar.
They added that there had been previous incidents at the address and described them as “allegations of a domestic nature”. Police also said that they had been interviewing witnesses and that they would oppose any bail application.
Miss Steenkamp’s publicist confirmed to the BBC that the 29-year-old model had died.
“Everyone who knew her is in tears. She was an absolute angel, the sweetest, sweetest human being, a kind human being,” Sarit Tomlinson said.
The news that Mr Pistorius may have killed his girlfriend will shock the country as the athlete is regarded as one of South Africa’s national icons, reports the BBC’s Peter Biles from Cape Town.
The incident is said to have taken place between 04:00 and 05:00 local time (02:00-03:00 GMT).
A police statement said that a woman was “fatally wounded” in the early hours of Thursday morning at a home a high-security compound in the Boschkop area of the capital.
The woman had been wounded in her head and upper body. Paramedics were at the house when police arrived, but she died at the scene. A 9mm pistol was recovered.
The athlete’s home is in the secure up-market Silver Woods gated compound on the outskirts of Pretoria.
Known as the “blade runner”, Mr Pistorius races wearing carbon fibre prosthetic blades after he was born without a fibula in both legs and had his legs amputated below the knee before his first birthday.
He reached the 400m semi-finals in the London 2012 Olympics. At the Paralympics he won silver in the T44 200m, gold in the 4×100 relay and gold in the T44 400m, setting a Paralympic record.
For years he dominated in his category at successive Paralympic Games.
In 2008 he won a legal battle over his blades with the International Association of Athletics Federations (IAAF) for the right to compete in able-bodied competitions.
His achievements have made him a “living legend” and placed him among global sporting royalty, South African broadcaster and commentator Daniel Silke told the BBC.
“He’s a household name, he’s a hero in the South African sporting context. He is of course someone who has overcome great difficulty and tragedy in his own personal life,” he added.
The South African Sports Confederation and Olympic Committee said it had been inundated with questions over the incident, but was in no position to comment.
Ed Warner, head of UK Athletics also reiterated that little was known about the incident but described the reports as “shocking and very sad,” adding that: “Pistorius is one of the great icons of Paralympic athletics but also of able-bodied athletics too”.
The International Paralympic Committee also declined to comment on the matter as a police investigation was under way.
South Africa has one of the highest rates of crime in the world and many residents keep weapons to protect themselves against intruders.
But gun ownership is strictly regulated and it is not easy to obtain a licence.
On Tuesday, a bill seeking to give police extra powers to arrest anyone carrying a dangerous weapon in public was tabled before parliament, following a spate of violent strikes and protests last year.
Reprinted from BBCnews