Probation officer and social worker Annette Vergeer told the court that Pistorius' legal team had told her about the payments, of 6,000 rand (about $540) a month, and she suggested that this showed remorse.
She said Pistorius had offered the family more besides the monthly payments. The athlete sold his car for about $35,000 and had that money given to the Steenkamp family, she said.
Pistorius' manager, Peet van Zyl, facilitated the monthly payments from Pistorius' bank account.
However, the Steenkamp family has rejected the money, the court heard.
Prosecutor Gerrie Nel said that Reeva's mother, June Steenkamp, had said she didn't want "blood money."
He told the court that Steenkamp's parents had said they would repay the monthly payments in full.
The Steenkamp family, through a lawyer, told Vergeer that it was "neutral" on the sentencing phase of the trial, Vergeer said.
She told the court that neither of Reeva Steenkamp's parents would testify for the state at the hearing.
Pistorius 'vulnerable' in prison
Judge Thokozile Masipa
, who presided over Pistorius' high-profile trial and delivered the verdict last month, will decide his sentence, which may or may not see him spend time behind bars.
The hearing, which began Monday, is likely to take several days, as the prosecution and defense put their arguments before the court.
Vergeer argued that Pistorius should not be sent to prison, saying that as a double amputee, he would be vulnerable to gang rape -- particularly if his prosthetic legs were taken away.
Nel suggested that if Pistorius were allowed to keep his prosthetic legs while in prison, he'd be fine.
But Vergeer said that even if the court allowed Pistorius to keep the legs, there was no guarantee the order would be complied with.
Pistorius was found guilty of culpable homicide, the South African term for unintentionally, but unlawfully, killing a person. However, the judge cleared him of murder in Steenkamp's death and granted him bail pending the sentencing hearing.
On Monday, a psychologist said Pistorius was a "broken man" after the death of his girlfriend, as his defense tried to convince a judge of his fragile emotional state.
Dr. Lore Hartzenberg said the Olympian described Steenkamp
as the love of his life. He lost faith in himself and his abilities after the killing, she said.
But the prosecution dismissed her findings.
"We're dealing with a broken man, but he's still alive," Nel said in an apparent reference to the fact that Steenkamp is not. "He has lost his love relationship ... but he will be able to love somebody else like that."
Sentence hard to predict
There is no legal minimum sentence for culpable homicide in South African law, so it will be up to the judge to decide.
A typical sentence is five to eight years. But it is a principle of South African law that the sentence should be tailored to the culprit as a whole person, as opposed to the crime. That makes predicting a sentence difficult, said Kelly Phelps, a CNN legal analyst.
The judge also found Pistorius guilty of one weapons-related charge involving a shooting at a restaurant.
The maximum penalty for that is five years behind bars. But he could get a lesser sentence, such as a fine or the loss of his gun license.