MILAN, Nov 21 (Reuters) - Telecom Italia (TIM) (TLIT.MI) said on Sunday it had received a 10.8 billion euros ($12 billion) approach from U.S. fund KKR (KKR.N) aimed at taking Italy"s biggest phone group private. KKR"s move comes as TIM"s CEO Luigi Gubitosi battles for survival after coming under fire from top investor Vivendi (VIV.PA) following two profit warnings in three months. TIM said KKR had set an indicative price of 0.505 euros for its possible buyout offer -- a 45.7% premium to the ordinary shares" closing price on Friday. KKR would also offer the same price for TIM"s savings shares . The TIM board, chaired by former Bank of Italy official Salvatore Rossi, met for several hours on Sunday afternoon but in a short statement it gave no indication of whether it would support the approach. It noted that KKR had termed its action as "friendly" and aimed at winning the backing of the company. Vivendi, which is pushing to replace Gubitosi, believes KKR"s offer does not adequately value TIM, a person close the French media group said. Vivendi faces a steep capital loss on its 24% TIM stake for which it has spent on average 1.071 euros a share. A new owner would also have to assume the group"s 29 billion euro gross debt. The government, which is preparing to tap billions of euros of European Union recovery funds to boost broadband connectivity in Italy, is aware of the need to find a way to shore up the former telecoms monopoly and protect its 42,500 domestic workers, sources have said. KKR"S CURRENT ROLE Gubitosi brought KKR onboard last year in a 1.8 billion euro deal that handed the New York-based fund a 37.5% stake in FiberCop, the unit holding TIM"s last-mile network connecting street cabinets to people"s homes. Vivendi sees Gubitosi as a short-term solution for TIM, people close to the matter have said. One person said on Sunday KKR"s plan may buy Gubitosi a few more months. read more TIM"s fixed line business is its most prized asset and is deemed strategic by Rome which can block any unwanted moves. Unable to stem TIM"s revenue haemorrhage, Gubitosi has started looking at ways to squeeze money out of TIM"s assets, revisiting in particular a plan to merge TIM"s fixed-line grid with that of fibre optic rival Open Fiber. Sponsored by the previous government, that project had run aground under Prime Minister Mario Draghi. KKR"s plan would see TIM carve out its fixed network to be run as a government-regulated asset along the model used by energy grid company Terna (TRN.MI) or gas grid firm Snam (SRG.MI), two sources close to the matter said earlier on Sunday. Separately private equity firms CVC and Advent have studied possible plans for TIM, working with former TIM CEO Marco Patuano, now a senior adviser to Nomura (7131.T) in Italy. A spokesperson for the two funds said they were open to working with all stakeholders on a solution to strengthen TIM, denying any contacts with Vivendi. A spokesperson for Vivendi said the French media group remained ready to work alongside Italy"s authorities and institutions for TIM"s long-term success. Vivendi had wanted TIM to retain control of the combined network assets in any Open Fiber deal. However, ceding control increasingly appears as the only way to overcome political and regulatory opposition to the single network plan. TAKEOVER SHIELD To oversee a strategic asset such as TIM"s fixed line, state investor CDP has taken a 9.8% stake becoming the group"s second-largest investor after Vivendi. KKR"s plan can only proceed with the government"s assent because Rome has special anti-takeover powers to shield companies deemed of strategic importance from foreign bids. TIM"s fixed network is also a key asset supporting the debt burden which was cut further below the investment grade level by credit rating agency S&P on Friday. read more TIM"s revenue have shrunk by a fifth over the past five years hit by aggressive competition at home from rivals such as Iliad, Vodafone (VOD.L), Wind Tre and Fastweb. ($1 = 0.8859 euros)