CDP, together with co-investors Macquarie and Blackstone, has been in talks since October with Atlantia to buy its 88% stake in toll-road unit Autostrade per l'Italia.

But no binding bid has yet emerged.

On Dec. 23, CDP sent Atlantia a new non-binding offer with an assurance it would proceed swiftly with due diligence but said it needed more time for analysis before a binding bid.

In a statement on Monday, Atlantia said the Dec. 23 offer had been below its board's expectations and below a previous range given by CDP itself.

"It (the offer) is still subject to further potential adjustments following the completion of due diligence process," Atlantia said.

The CDP-led consortium valued Autostrade at 8.5-9.5 billion euros ($10.4-11.6 billion) in an earlier preliminary offer, but potential legal claims and fallout from a probe has since made it more cautious.

A source close to the matter said on Monday the CDP offer on Dec. 23 valued the whole of Autostrade at roughly 8 billion euros.

A deal between the two parties would put an end to a dispute sparked by the 2018 collapse of a motorway bridge operated by Autostrade that killed 43 people.

"The board confirmed to the CDP consortium its willingness to evaluate a potential binding offer for its entire stake held in Autostrade," Atlantia said.

Atlantia, which is controlled by the Benetton family, is in the meantime also pressing ahead with an alternative plan to spin off and sell or list its 88% stake.

CDP Equity, which is involved in the bid, declined to comment.

($1 = 0.8194 euros)

(Reporting by Stephen Jewkes; Editing by Andrea Mandala and Andrew Cawthorne)