* Elliott ups potential stake to 13.7 pct
* AGM on April 24 set to vote on Elliott's request
* TIM board decides Monday whether to take action
* Three proxy advisers recommend backing Elliott proposal (Recasts after Elliott builds stake, details plan for TIM)
MILAN, April 9 (Reuters) - Activist fund Elliott has raised its interest in Telecom Italia (TIM) as it gears up for a leadership showdown later this month with top investor Vivendi.
Elliott, which revealed on Monday it had a potential TIM stake of 13.7 percent, and Vivendi will face off for the first time on April 24 when shareholders will be asked to vote on the fund's proposal to replace some board members nominated by the French media group with Italian business leaders.
Paul Singer's fund called in March for six Vivendi-nominated board members, including TIM Chairman and Vivendi CEO Arnaud de Puyfontaine, to be replaced to improve governance and strategy.
TIM has lost a quarter of its market value since Vivendi first took a stake in mid-2015 and increasingly tightened its grip on Italy's biggest telecoms group.
Three proxy advisers, including Glass Lewis and ISS, have recommended shareholders back Elliott's board proposal.
The fund has said it would push a more independent board to convert TIM's savings shares, return to paying dividends and work on strategic moves such as a listing and partial stake of TIM's network to cut debt and unlock value.
"Vivendi appears to be more of a liability than an asset for TIM," ISS said in a report on Monday. "TIM has had its fair share of management and board turnover over the last few years, and Vivendi's influence has not brought about stability."
Elliott and its affiliate funds hold around 8.8 percent of TIM's ordinary shares, a SEC filing showed, compared with Vivendi's 23.9 percent. The activist investor also owns call options which amount to an additional potential stake of 4.9 percent stake, according to Reuters calculations.
However, even if Elliott's candidates win shareholder backing at TIM's annual general meeting, no clear indication of who will end up calling the shots at Italy's biggest telecoms group is likely to emerge for several weeks.
Following Elliott's move last month, eight board members nominated by Vivendi resigned, triggering a vote on full board renewal at a separate shareholder meeting called for May 4.
Elliott told investors in an online presentation on Monday it was in favor of TIM's existing management team, including CEO Amos Genish and his business plan. However, Genish may not want to remain if he cannot count on a majority backing from TIM's board, sources close to the matter have said.
Outlining some of its other plans for TIM, Elliott said it could sell down further its stake in tower unit INWIT, consider a tie-up with a local company for its majority-owned Brazilian unit and separate its network assets.
The latter could unlock 7 billion euros in hidden value, the fund said, and drive a re-rating of shares.
Elliott believes there will be no need to choose a new board in May if its candidates are elected in April. TIM, however, says even if Elliott's candidates are elected, a new board would have to be voted on in May.
TIM's board is meeting on Monday to decide whether to take action against a decision by its statutory auditors to include Elliott's proposal in the April 24 meeting agenda. Sources said TIM may ask a judge to rule on the matter.
All three proxy advisers recommended shareholders vote in favor of Genish, who was co-opted by the board in September after the resignation of his predecessor. Shareholders need to approve his appointment for him to continue beyond the AGM.
Most institutional investors approve of Genish, despite him being an ally of Vivendi Chairman Vincent Bollore, because of his track record as a telecoms veteran and dealmaker in Brazil.
Genish already features in the slate put forward by Vivendi.
This may prove a conundrum for some investors who may want to weaken Vivendi's grip but will not want to lose Genish, whose three-year business plan presented last month was welcomed.
After a volatile start, shares in Telecom Italia closed up 0.1 percent, compared with a 0.5 percent rise in Milan's blue-chip index. Around 2.6 percent of TIM's capital traded on Monday. (Additional reporting by Valentina Za and Stefano Rebaudo Editing by Alexander Smith)