Modi's office issued the rebuttal shortly after Rajnath Singh gave an emotional statement to the media, saying he would quit public life if reports in recent days about his son, Pankaj Singh, proved to be true.
"The reports are plain lies," the prime minister's office said in a statement, adding that they "constitute a malicious attempt at character assassination and tarnishing the image of this government."
Earlier, a newspaper report said that Rajnath Singh had complained to leaders of the ruling party that a ministerial colleague had been spreading false rumours against him. The report cited an unidentified senior source saying there was a "power struggle" in the party.
Singh, a former president of the ruling Bharatiya Janata Party (BJP), holds the second most senior cabinet post after Modi and is responsible for policing and internal security.
Modi has cultivated an austere image since ousting a Congress-led government tarnished by corruption scandals that affected industries ranging from coal to telecoms.
The 63-year-old prime minister has installed close ally and campaign manager Amit Shah as BJP president as he seeks to strengthen his hold on power ahead of state elections that could boost its presence in parliament's upper house.
Although the Hindu nationalist BJP won the biggest lower house mandate in 30 years in May, its lack of an upper house majority has already delayed legislative initiatives such as allowing greater foreign investment in the insurance industry.