(Bloomberg) -- KKR & Co. raised its bid for Telecom Italia SpA’s landline network, giving the US private equity firm an edge over its rival, the Italian state lender Cassa Depositi e Prestiti SpA, according to people familiar with the matter.

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While a new offer from Cassa Depositi’s consortium is basically unchanged from its previous bid, KKR has improved its terms by €300 million to €400 million, possibly making it more appealing, said the people, who asked not to be identified discussing the private deliberations.

Still, KKR’s sweeteners could in theory be worth about €2 billion, if a complex series of conditions are met involving services agreement contracts and changes to the network unit’s debt structure, the people added. If the conditions are achieved, KKR’s bid could reach about €23 billion from the previous offer of €21 billion, which already included about €2 billion in a performance-based “earnout.”

Cassa Depositi’s bid was around €19.3 billion, people familiar with the matter have told Bloomberg.

Still, despite the increase, KKR’s proposal remains far below the estimated €30 billion sought by Vivendi SE, Telecom Italia’s largest shareholder. Representative for Telecom Italia, Cassa Depositi and KKR all declined to comment.

The offers reported in the press still seem very poor and the season for new bids for Telecom Italia’s network is definitely closed, people familiar with Vivendi’s position said. It’s necessary to open a new chapter and explore fresh options that could create more industrial value for the company, they said.

KKR’s bid is free of any major competition authority risks, while Cassa Depositi would likely face antitrust issue that could arise based on a series of cross holdings. The Italian state lender owns nearly 10% of Telecom Italia and controls the carrier’s smaller rival, Open Fiber SpA.

Possible remedies to the antitrust issues could include selling some Open Fiber assets in metropolitan areas such as Milan and Rome, people familiar with the matter said earlier.

The sale of Telecom Italia’s most valuable asset has become more urgent in the last year amid mounting pressure to slash its €30 billion in gross debt as interest rates rise.

Telecom Italia’s board is divided between entering exclusive talks with one of the bidders or pulling the plug on the sale since neither offer is likely to match Vivendi’s expectations. The board will meet on June 19 and June 22 to discuss the offers, Telecom Italia said on Friday, confirming an earlier Bloomberg report.

(Updates with details on Vivendi’s position in sixth paragraph)

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