Police in Pretoria said they would ask at a hearing scheduled for Friday that Pistorius be held without bail, The New York Times reported. They refused to speculate on a motive for the shooting but spokeswoman Brig. Denise Beukes said officers had been to Pistorius' home previously because of problems of a "domestic nature."
Model Reeva Steenkamp, 30, was shot in the head and arm around 3 a.m. and declared dead at the scene, police said. Some South African newspapers suggested Pistorius might have thought Steenkamp was a burglar.
Pistorius, 26, appeared severely shocked when questioned by police at his home in Pretoria, the Afrikaans daily Beeld reported. Police said they found a 9mm pistol in the house.
Henke Pistorius told the South African Broadcasting Corp. he would not comment on his son's arrest.
"I don't know nothing. It will be extremely obnoxious and rude to speculate," he said. "I don't know the facts. If anyone makes a statement, it will be Oscar. He's sad at the moment."
Steenkamp and Pistorius had a "healthy, fabulous" relationship, Steenkamp's publicist, Sarit Tomlinson, told Sky News, the Times said.
Dozens of people posted messages on Steenkamp's Facebook page expressing sadness and shock.
"Rest in peace beautiful soul," read one message. "Such a terrible tragedy and loss," read another.
The Times said Steenkamp sent a Valentine-themed tweet Wednesday: "What do you have up your sleeve for your love tomorrow???"
Pistorius was born without a fibula in both legs, which were amputated below the knee when he was 11 months old. He races wearing carbon-fiber prosthetic blades and is known as the "Blade Runner."
Pistorius is the first double amputee to run in the Olympic Games, qualifying for the 400-meter semifinals and 4x400-meter final at the 2012 Summer Olympics. He also competed in the Paralympics, where he won gold medals in the 400 meters and in a relay and silver in the 200 meters.