Transport and Infrastructure Minister Maurizio Lupi was already resisting pressure to resign over revelations that one of the businessmen arrested in the case had given his son, Luca Lupi, a €10,000 Rolex watch.
Examining magistrates suspect that the same businessman, Stefano Perotti, organized a job for the recently-graduated son at an engineering firm run by his brother-in-law, at the request of the minister.
Lupi on Tuesday denied making any such request, saying he had never sought favours for his children and that to do so "would have been a grave error and, I presume, a crime."
It emerged on Wednesday that wiretap evidence in the case includes a recording of Lupi calling a senior official in his ministry in June 2014 and telling him: "You must come and meet my son."
Further wiretaps indicate that the official, Ettore Incalza, asked Perotti to arrange things and, by the end of the month, Luca Lupi had his job, media reports said.
Incalza, who was arrested on Monday, was the top official in charge of major public works in Lupi's ministry until the end of last year, when he left to become a consultant.
The minister has described Perotti as a family friend and the wiretaps suggest he was also close to another businessman, Francesco Cavallo, who has also been arrested in the kickbacks case. Several recordings reveal that Cavallo regularly introduced himself to business contacts as "Lupi's man."
Prime Minister Matteo Renzi was widely reported to have told Lupi to resign to restrict the damage the case is doing to the government's credibility.
But commentators say the centre-left premier cannot simply sack Lupi because he is a leading member of the New Centre-Right (NCD), the junior partner in a governing coalition dominated by Renzi's Democratic Party (PD).
With tough parliamentary tests looming over reforms of the legislature and the electoral system, Renzi can ill-afford to lose the backing of the smaller party.
Contracts worth €25 billion
Lupi was due to make a statement later in the day to the Chamber of Deputies, where opposition parties have tabled a no-confidence motion aimed at forcing him to resign.
Incalza, his associate Sandro Pacella, Perotti and Cavallo were all arrested on Monday on suspicion of systematically rigging more than €25 billion worth of tenders for major public works.
The judges claim that the organized corruption allowed the men to skim off around one percent of the value of contracts for high-speed rail links, new metro lines in Rome and Milan and other huge projects.
The scandal is the biggest of its kind since the "tangentopoli" cases of the early 1990s.
That episode led to the indictment of half the country's lawmakers but, two decades later, Italy is still struggling to shake off a reputation for corruption that economists say has cost the country billions in lost investment.