The political storm engulfing Brazil’s far-right president, Jair Bolsonaro, has intensified with reports that federal police investigators have identified his son as one of the alleged key members of a “criminal fake news racket” engaged in threatening and defaming Brazilian authorities.

One of Brazil’s top newspapers, the Folha de São Paulo, claimed an investigation by Brazil’s equivalent to the FBI had homed in on Carlos Bolsonaro, the president’s social-media-savvy son.

Carlos Bolsonaro, 37, rejected the claims as “garbage” and “a joke” on Twitter, where he has 1.7 million followers.

But the allegations will deepen the crisis consuming Bolsonaro’s 16-month-old government and further distract from the country’s efforts to tackle the coronavirus pandemic, which has killed more than 4,000 Brazilians.

Bolsonaro’s administration was already floundering before the recent resignation of his powerful justice minister, Sergio Moro, with another of the president’s sons, Flávio Bolsonaro, facing police scrutiny for suspected corruption and ties to Rio de Janeiro’s mafia.

Bolsonaro’s dismissive reaction to the coronavirus crisis has sparked outrage across the political spectrum and pot-banging protests.

The latest political melodrama exploded on Friday when Moro resigned and publicly accused the president of attempting to improperly meddle in the operations of the federal police by sacking the federal police director, Maurício Valeixo, that morning.

Carlos Bolsonaro called the claims a ‘joke’. Photograph: Sergio Lima/AFP via Getty Images

Moro alleged that Bolsonaro, for reasons that remain unclear, hoped to replace Valeixo with someone more amenable to discussing federal police business and sharing intelligence reports with the president.

The man widely touted as Valeixo’s successor is Alexandre Ramagem, the head of Brazil’s intelligence agency, who is reportedly a friend of Carlos Bolsonaro’s.

A photograph of Ramagem and Carlos Bolsonaro fraternising at a New Year’s Eve party was published by Brazilian media on Saturday.

The Folha de São Paulo claimed federal police officials were convinced Bolsonaro had fired Valeixo “because he was aware the corporation was closing in on his son”.

An editorial in another leading broadsheet, O Globo, said Bolsonaro seemed to want “to turn the federal police into a personal police force, as if he were some kind of third world dictator”.

The Folha de São Paulo claimed another of Bolsonaro’s sons, the congressman Eduardo Bolsonaro, was also being investigated for possible involvement in the fake news group. Eduardo Bolsonaro, who is the South America representative of Steve Bannon’s far-right group The Movement, claimed on Twitter the claims were “an attempt to confuse the population”. He also retweeted an image of a dog defecating on the newspaper’s masthead.

Jair Bolsonaro and his backers have been battling to regain control of the narrative since Moro’s damaging exit, painting the former judge as a pro-choice, anti-gun traitor to their far-right cause.

Bolsonaro denied any wrongdoing in a rambling speech in which he claimed he was the victim of a political conspiracy. “Powerful people have risen up against me. This is a reality. This is the truth. I’m fighting against the system – against the establisher [sic],” he claimed.

“We are, in the figurative sense, going to get shot in the face loads. But we will fulfil our mission,” Bolsonaro vowed.

But claims investigators had identified the president’s son as part of the alleged fake news network – something political foes have long suspected – prompted fresh calls for the president’s impeachment and reinforced growing doubts over whether the rightwing populist would complete his four-year term.

“The government is crumbling,” said Eliane Cantanhêde, a political commentator for the conservative Estado de São Paulo newspaper.

“I wouldn’t bet on an impeachment – either for or against,” she added, pointing to reluctance among some politicians to impeach a third president in less than three decades. “But it’s also hard to imagine Bolsonaro making it through nearly three more years in the midst of this chaos. It’s a real pickle.”

Marcelo Freixo, a leftwing congressman, said he would take legal action to block Ramagem’s appointment.

“Bolsonaro wants to control the federal police ... because he doesn’t want the federal police to advance in its investigations related to his sons. There’s no doubt about it,” Freixo claimed. “Bolsonaro is protecting his children.”

The Guardian has approached Brazil’s presidential communication secretariat for comment.