The request to shareholders is part of a ruthless fight over the Italian company's future ahead of its annual general meeting next month.
Elliott, which says that it has a five percent holding in Telecom Italia (TIM), has criticized largest shareholder Vivendi's management.
Now the firm has asked for a shake-up at the top, "to improve both governance and performance at TIM".
"Elliott believes that the company is managed in the interest of Vivendi and to the detriment of all other TIM shareholders," it wrote.
"Yet, Vivendi controls only 24 percent of the ordinary shares and its stake represents only 18 percent of total equity value."
Among the board members Elliott wants to remove is executive chairman Arnaud Roy de Puyfontaine, also Vivendi's CEO, when the AGM is held on April 24th.
Sometimes called a "vulture" fund, Elliott has regularly invested in companies in difficulty or whose shares it considers undervalued, and often engages in showdowns with those companies' management.
Vivendi, owned by French billionaire Vincent Bollore, responded on Friday by saying it will examine Elliott's statement with an "open mind" but called the firm "a hedge fund well known for its short-termist initiatives".
"It is not sure that the plan to dismantle the group and destabilize the teams will create value," it said in a statement.
Vivendi might have a minority holding in TIM but has de facto control of the company thanks to its stranglehold on board and executive positions.
TIM's current CEO Amos Genish, appointed last year, is close to Bollore. Genish is TIM's third CEO is as many years.
Michel Sibony, appointed as Head of the Procurement Unit and Real Estate Department last week, holds several functions within the Bollore Group and Havas, says Elliott. He was named Chief Value Officer at Vivendi just before Christmas.
Elliott's suggested six board replacements include Fulvio Conti, former CEO of utility multinational Enel and Extraordinary Commissioner of airline Alitalia Luigi Gubitosi, who between 2007 to 2011 was CEO of fellow Italian telecommunications company Wind.
"Elliott believes that these six candidates can empower the board to correct the persistent undervaluation that is undeniably present at TIM," it said.
Earlier this month TIM announced largely positive results for 2017, with group EBITDA, a measure of operating performance, up nearly five percent year-on-year to €8.7 billion and revenues up nearly three percent to just under €20 billion.
However, Elliott says that TIM's stock "remains highly undervalued notwithstanding improving fundamentals" and that the growth last year came after "years of persistent declines."
It highlighted the 35 percent drop in the value of the company's share price from when who it calls "Vivendi nominees" joined the board in December 2015 to "the day before our interest in the company was made public".
It cites several instances of Vivendi "exercizing its control without regard for minority shareholders' divergent interests".
Among them was TIM's January 2017 awarding of an advertising contract to Havas, which is owned by Vivendi, worth a rumoured €100 million.
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