Britain's Sunday People newspaper quoted the father as saying: "We want the world to know her real name."
But Indian media later said the father denied giving consent for her name to be revealed.
Indian law protects sex crime victims by prohibiting identification.
One minister, Shashi Tharoor, has urged authorities to reveal the name so it can be used for a new anti-rape law.
The woman, 23, died last weekend in a hospital in Singapore from injuries suffered during last month's attack.
The father was quoted as telling the Sunday People: "My daughter didn't do anything wrong, she died while protecting herself.
"I am proud of her. Revealing her name will give courage to other women who have survived these attacks. They will find strength from my daughter."
However, the Hindustan Times later quoted the father as saying he would want the name made public only if a law was named after her.
"I have only said we won't have any objection if the government uses my daughter's name for a new law for crime against women that is more stringent and better framed that the existing one," the Indian paper quoted him as saying.
The laws on identification were introduced to protect victims from the social stigma associated with rape, and apply even after the death of the victim.
Although the system has not always been watertight, the high-profile case has brought a tough stance from the authorities.
Police filed a case against broadcaster Zee News after it carried an interview with the friend who was with the victim during the attack.
The victim's friend was not named but his face was shown and police are investigating whether Zee News broke broadcasting laws relating to disclosure of the victim's identity.
However, it remains unclear what could be done if the father chose to publicly name his daughter.
Last week, Mr Tharoor, the junior education minister, called on the authorities to reveal the name of the gang-rape victim so that the new anti-rape law could be named after her.
He wrote: "Unless her parents object, she should be honoured and the revised anti-rape law named after her. She was a human being with a name, not just a symbol."
The Sunday People said the father had given it permission to name him and his daughter.
It carried a photograph of the father but said the family had requested no photograph of the victim be used.
In the interview, the father also renewed his calls for the men who carried out the attack to be hanged.
"Death for all six of them. These men are beasts. They should be made an example of and that society will not allow such things to happen," he said.
Five men have been charged with abduction, rape and murder. A sixth suspect is expected to be tried as a juvenile.
A pre-court hearing for the five was held in the Saket area of the Indian capital on Saturday and the men have been summoned to appear in court on Monday.
In his interview with Zee News, the friend said he and the rape victim had boarded a bus after a trip to the cinema and after failing to flag down an auto-rickshaw.
He said the bus had tinted windows, and that he believed the group of men had laid a trap for them.
He confirmed that the assailants had later thrown them off the bus and tried to run them over.
The case has caused a national outcry, and there have been frequent protests calling for greater protection for women.
The BBC's Andrew North, in Delhi, says it continues to put Indian life under a sharp magnifying glass and for many people it is uncomfortable viewing.
The father of an Indian woman who was gang raped in Delhi and later died says her name should be made public so she can serve as an inspiration to other sex crime victims, a UK paper reports.