MILAN, June 8 (Reuters) - U.S. fund KKR and a rival consortium comprising state lender CDP and Australian fund Macquarie are readying to submit slightly improved bids for Telecom Italia's (TIM) network ahead of a June 9 deadline, sources said.

Debt-crippled TIM is seeking improved offers for its most valuable asset after having assessed as not yet adequate the proposals received in May.

KKR and the CDP-led consortium offered 21 billion euros($22.5 billion) and 19 billion euros, respectively, for TIM's domestic landline grid and its submarine unit Sparkle, sources have previously said. KKR's approach included a performance linked outlay worth 2 billion euros.

Both contenders are expected to present some limited improvements compared with their previous bids, three sources familiar with the matter said.

TIM directors are due to meet to review the proposals on June 19 and to give a response on June 22, two separate sources told Reuters.

Treasury-owned CDP is the second-largest investor in TIM with a 10% stake after France's Vivendi.

The state lender and Macquarie are also co-investors in TIM's smaller rival Open Fiber.

Their approach for TIM's grid is part of a plan to combine Open Fiber with the former phone monopoly infrastructure, which has been complicated by antitrust issues.

People familiar with the matter told Reuters last month the Treasury would be ready to welcome CDP joining forces with KKR for TIM's grid, with one source saying government officials were pushing for CDP to drop its separate proposal.

But Macquarie is raising legal hurdles about a possible alliance between CDP and KKR, other sources said at the time, citing clauses in the Open Fiber shareholder pact.

The sale of Telecom Italia's infrastructure and its submarine cable unit Sparkle is a focal point of CEO Pietro Labriola's plan to slash the former phone monopoly 26 billion euro debt pile and turnaround the battered telecoms group.

Such a strategy is challenged by Vivendi which is seeking a 31 billion euro valuation to back a sale.

($1 = 0.9323 euros) (Reporting by Elvira Pollina; editing by David Evans)