The RTS report is based on unnamed sources and Swiss authorities reached Sunday would not confirm or deny it.
Riadh Ben Aissa was arrested in Switzerland and is jail on suspicion of corrupting a public official, fraud and money laundering tied to his dealings in North Africa.
A review by the Montreal-based engineering and construction giant led to it finding $56 million of payments to unidentified foreign agents.
The RTS report says the allegations against Ben Aissa tie him to at least $139 million in payments. The report says authorities have tracked money moving from SNC to Swiss bank accounts registered to companies in the British Virgin Islands.
The report further cites sources saying some of the money then ended up in accounts that Ben Aissa controlled.
A spokeswoman for the Attorney General of Switzerland would neither confirm nor deny the report.
"According to the procedure in place in Switzerland, the criminal investigation before the trial phase is not public," Jeannette Balmer said in an email exchange.
"Therefore no documents are published during this phase of procedure."
Ben Aissa was in charge of business dealings in his native Tunisia as well as Libya, where the company won lucrative contracts with the former regime of Moammar Gadhafi.
The company has insisted that none of the $56 million in payments were directed to Libya.
The RCMP executed search warrants at SNC-Lavalin's (TSX:SNC) Montreal headquarters in April. The company said it was co-operating with the RCMP investigation.
The raid followed an investigation into bidding on projects in Bangladesh that prompted RCMP searches of SNC's Toronto-area offices last September.
Earlier this month, Ben Aissa's brother filed a $5-million lawsuit in Quebec Superior Court alleging the company caused him harm by using Ben Aissa as a "scapegoat" while protecting its interests in Libya in the face of political change.
Rafik Benaissa, an American-based orthopedic surgeon, said the Canadian company's attack on his brother followed 27 years of supporting and funding his work with the regime of former leader Moammar Gadhafi.
The allegations have not been tested in court.