(Adds TIM share performance, details)
MILAN, April 27 (Reuters) - Activist fund Elliott stepped up pressure on Telecom Italia (TIM) shareholders to change the group's board at a May 4 vote, in a move to reduce Vivendi's influence and remove "an impediment to TIM's full potential".
The two groups have engaged in a fierce war of words since Elliott last month became TIM's second-largest investor, with a stake of 9 percent.
Elliott is pushing to shake-up the way top investor Vivendi, which holds a 24 percent stake, runs the former state monopoly.
"Put plainly, Vivendi's repeated missteps have eroded the value of your investment," Elliott said in a letter to TIM shareholders dated April 26.
The fund has repeatedly said that TIM's share price, strategy and valuation had suffered under Vivendi's leadership and accused the French media group of only serving its own interests.
TIM has lost a quarter of its market value since Vivendi took a stake in the group in mid-2015.
The company's share price is down 4 percent since May 2017, when Vivendi tightened its grip on TIM by appointing two thirds of the group's board.
In the letter Elliott also said it was "deeply troubled" by the recent corruption allegations involving Vivendi's top investor Vincent Bollore.
The French billionaire tycoon was placed under formal investigations over allegations his company, Groupe Bollore , undercharged for work on behalf of presidential candidates in two African nations in return for port contracts.
"With proper stewardship at the board level, and a highly qualified independent board of directors, TIM's prospects are bright," the fund wrote in the document.
Elliott reiterated its support for TIM Chief Executive Amos Genish, his management team and his business plan and added that a "refreshed, highly qualified, independent board" could ensure the CEO then has the support needed to execute the new strategy.
Vivendi has said Elliott's vote in favour of Genish at an April 24 AGM showed its "inconsistency". Beyond its push for a renewal of the board, Elliott has proposed selling a majority stake in TIM's soon-to-be created network company, NetCo, and merging it with local rival Open Fiber. It also said TIM's Brazilian business could be combined with a local peer.
The market has cheered Elliott's plans, with shares in the telecoms group rising almost 14 percent since the fund first disclosed a holding in TIM.
New York-headquartered Elliott Management Corp was founded by U.S. billionaire Paul Singer and manages two funds with combined assets under management of about $35 billion. Its push for change at Telecom Italia is being spearheaded by London-based affiliate Elliott Advisors (UK).
Reporting by Giulia Segreti; editing by Jason Neely