TIM directors gathered on Thursday to draw a response to rival non binding approaches for its landline grid by KKR (KKR.N) and a consortium comprising state lender CDP and fund Macquarie (MQG.AX), worth 21 billion euros ($23 billion) and 19.3 billion euros, respectively.
In a statement after the board meeting, TIM said it deemed the bids "not yet adequate", adding at least one of the bidders had expressed its readiness to improve its non-binding offer.
Three sources with knowledge of the matter said KKR was willing to work further on its bid's terms.
Italian last month said her right-wing administration would not intervene at this stage in the network sale process, although it is on alert to avoid any risk to the national interest.
($1 = 0.9081 euros)Reporting by Elvira Pollina and Juby Babu, Editing by Franklin PaulOur Standards: The Thomson Reuters Trust Principles.