MILAN - U.S. fund KKR strengthened its lead in the race to secure the landline grid of Telecom Italia (TIM) when it offered to raise its bid by up to or over 2 billion euros ($2.2 billion), two people with knowledge of the matter said.

The value of KKR's offer could top 23 billion euros overall, widening the gap with a rival proposal by a consortium comprising Italian state lender CDP and Australian fund Macquarie.

The improved offer would still be short of a valuation of more than 30 billion euros for the grid sought by TIM's top shareholder Vivendi. But it remains TIM CEO Pietro Labriola's best option to pull off plans to rescue the debt-laden firm via a sale of the network.

Sources had previously told Reuters that both Labriola and some leading Italian officials already saw KKR as the strongest bidder prior to Friday's proposal.

TIM said late on Friday it had received two new offers for its grid, without providing details. It had sought improved offers for its most valuable asset after having assessed as not yet adequate the proposals received in May.

Labriola plans to focus TIM's efforts on its ServCo services business and sell its NetCo unit comprising the domestic fixed-access network and international submarine cable unit Sparkle to cut debt.

The overall value of KKR's bid hinges on the terms of the contracts linking ServCo to NetCo, the sources said, adding that the fund had asked for four weeks to carry out due diligence to discuss such conditions.

KKR declined to comment.

The source said KKR was also offering to leave TIM a stake in NetCo, having previously indicated it would also welcome a state entity as a shareholder in the grid, which is Italy's main telecoms infrastructure.

CDP and Macquarie have kept the economic value of their proposal little changed, suggesting a number of measures to address antitrust issues, another two people close to the matter said.

The antitrust problems are linked to the fact that CDP and Macquarie also own fibre-optic wholesale provider Open Fiber. One remedy could involve parts of Open Fiber being sold off.

CDP is also TIM's second-largest shareholder with a 10% stake.

TIM's board meets to review the proposals on June 19 and is expected to take a decision on June 22. ($1 = 0.9305 euros)

(Reporting by Akriti Sharma in Bengaluru and Elvira Pollina in Milan; Editing by Marguerita Choy and Leslie Adler)