Investigators opened a probe last year into the 2013 failure of the Chil Srl newspaper distribution and advertising firm, founded by Tiziano Renzi, the daily Il Secolo XIX reported.
The newspaper said prosecutors have asked for an extension of the investigation into Renzi senior and two others. The public prosecutor's office was not immediately available for comment.
Should the case go to trial it would prove a big embarrassment for fresh-faced Renzi junior, who took over in Italy earlier this year on a pledge to clean up politics and tackle widespread corruption.