MILAN, Jan 20 (Reuters) - Italy's biggest phone group, Telecom Italia (TIM), on Wednesday started work on a plan that could keep Chief Executive Luigi Gubitosi in his role when it names a new board this year, two sources familiar with the matter said.

The group's outgoing board, whose three-year term is coming to an end, started discussions to draft a slate of board nominees which it must file at the beginning of March, in time for shareholders to vote upon on March 31, the group said.

TIM has picked headhunter Egon Zehnder to help with the board's selection process.

The slate of nominees is part of a plan supported by Italy's government, which holds 10% in the group through Cassa Depositi e Prestiti (CDP), and French media giant Vivendi, which is TIM's top investor with a 24% stake.

"The idea is that the board would put forward its own list and Gubitosi's name would be on it," one of the two sources familiar with the matter said.

Vivendi's support is crucial for the plan to succeed.

Relations between the French group and Italy have been hurt by Rome's recent decision to pass a law that may curb Vivendi's interests in the country and strengthen the hand of broadcaster Mediaset in a long-running legal battle against Vivendi.

However, all the board members, including Vivendi's representatives, agreed with the board's decision to present its own list of candidates, TIM said.

Gubitosi took the helm at TIM three years ago after activist fund Elliott won a governance battle against Vivendi, succeeding in appointing 10 directors out of 15 to the board.

He has has focused on shedding assets to cut debt, while working on combining TIM's fixed network assets with those of state backed rival Open Fiber.

Gubitosi's appointment is part of a broader compromise the government is discussing with Vivendi as they attempt to heal relations and press on with the single-network project, sources have said previously. (Reporting by Elvira Pollina and Giuseppe Fonte Editing by Keith Weir and John Stonestreet)