Bloomberg sources indicate that the bid will be increased by around €2 billion, valuing the network at around €20 billion in total.
This, however, is still well short of the €31 billion figure reportedly being sought by TIM's biggest investor Vivendi.
TIM is reportedly eyeing a sale of the network to allow it to cut its €25 billion debt pile.
Earlier this month, CDP called a board meeting to approve a long-delayed offer for the fixed network. the state-backed firm also needed government approval for the bid.
TIM had previously called on both CDP and KKR to outline fresh proposals for the division by an April 18 deadline.
CDP already has a 10% stake in TIM and previously teamed up with the Australian infrastructure fund Macquarie in a bid for the network.
Its main rival is a bid from US investment firm KKR, which previously purchased a stake in TIM’s fixed network.
TIM said last month that the offers tabled for its network grid by KKR did “not wholly reflect the value of the asset”.
TIM has previously added that it aims to facilitate an improved offer from the parties in the coming weeks.
