Championed by the previous government of Mario Draghi, the multi-billion-euro bid is part of a broader plan to combine TIM's network assets with those of smaller rival Open Fiber to create a unified broadband champion under CDP's control.
Due by this Wednesday, Nov. 30, an offer is also central to TIM CEO Pietro Labriola's plans to break-up the struggling phone group and cut its 25 billion euro ($26 billion) debt.
But key officials in Giorgia Meloni's right-wing government have expressed strong reservations over the plan, raising doubts over whether CDP will be able to submit its offer, sources told Reuters on Friday.
A government source did not rule out Meloni's decision on TIM could come as early as Monday.
The prime minister is also due to hold talks on Monday with union officials from TIM, which has more than 40,000 staff.
Marking a break with the past, Meloni on Friday entrusted the government's broadband strategy to Cabinet Undersecretary Alessio Butti, who has openly criticised CDP's plans for TIM.
Butti has called instead on CDP to take over cash-bleeding TIM in full to then sell its service operations, including its Brazil-listed unit.
Fresh uncertainty over its future is hitting TIM at a time when rising rates increase the drain on the group's cash flow from interest payments.
TIM shares fell as much as 4% in early trade, but by 1040 GMT they had reversed losses and were up slightly.
"How the government intends to reach its stated objective to have a (single) telecommunications grid in public hands remains unclear," Intesa Sanpaolo wrote in a research note.
"The timing to find a path starts to narrow considering that TIM's available liquidity covers debt maturities until mid-2024 and debt refinancing looks tougher than in the past."
($1 = 0.9549 euros)
(Reporting by Elvira Pollina and Giuseppe Fonte; writing by Valentina Za; Editing by Keith Weir)