MILAN, Dec 6 (Reuters) - Shares in Telecom Italia (TIM) fell on Monday hit by growing uncertainty over a $37 billion takeover approach by U.S. fund KKR.

After receiving KKR's proposal on Nov. 19 and losing its fourth chief executive in six years a week later, TIM has failed to make progress in relation to what would be Europe's biggest ever private equity buyout deal.

Sources have said the new governance set up is precarious and hinders decision-making.

After CEO Luigi Gubitosi lost out in a clash with TIM's top investor Vivendi veteran manager Pietro Labriola was promoted general manager but a new CEO cannot be named until Gubitosi or another director vacates a board seat.

Vivendi, which declined to comment on Monday, said on Sunday it was open to handing the state control of TIM's fixed network if the group parted with its most prized asset. In the past it had demanded TIM retain majority ownership.

"We understand this change of opinion as a way of hindering KKR's path," Caixa analysts said. Further muddying the waters, Italian press at the weekend mentioned a meeting between Vivendi and state-owned investor CDP, TIM's second-largest shareholder, over a possible alternative plan to KKR's.

Two sources close to the matter told Reuters Vivendi was trying to build an alliance with CDP.

"A "plan B" in alternative to the current, or an increased, bid by KKR ... could suggest political interference, increased uncertainty ... and further frustration for minority shareholders," Banca Akros analyst Andrea Devita said.

Shares in TIM fell 2.3% by 0913 GMT against a rising Milan all-share market, after dropping 4.4% earlier to 0.444 euros/share and triggering a trading suspension.

Vivendi faces a large capital loss on its 24% TIM stake at the 0.505 euros/share KKR offered. The U.S. fund has said the price is only indicative and requested access to TIM's books before a formal offer.

TIM however is yet to open a data room and the special committee it has set up to deal with the KKR's bid failed to reach a decision over advisers at the weekend. ($1 = 0.8863 euros) (Reporting by Elvira Pollina and Andrea Mandala; writing by Valentina Za)