The French media group, which owns 24 percent of TIM, has been at loggerheads with Elliott since the U.S. hedge fund took a stake in Italy's biggest phone group earlier this year and ended up wrestling board control away from Vivendi in May.
The battle of words has intensified in recent months as TIM fell under pressure due to tougher competition at home, which in turn prompted speculation that Genish's position at the company could be at risk.
Shares in TIM are down around 30 percent so far in 2018, worse than a 15 percent fall in the European telecoms index <.SXKP>.
"Vivendi supports TIM's CEO Amos Genish unreservedly and values his contribution," Vivendi said on Saturday, confirming comments also reported by Italian media on Saturday.
"We believe that this vicious rumour campaign is orchestrated by Elliott, whose lack of organisation is bringing down Telecom Italia," added Vivendi.
The attack follows a similar move last month when Vivendi said TIM's performance had been "disastrous" since Elliott seized control of the Italian firm's board.
When contacted by Reuters on Saturday, Elliott referred to the statement it had released in response to Vivendi's previous accusations in September.
At the time, Elliott said Vivendi had "fallen prey to the 'short-termism' it has previously decried", casting judgement on the new board just four months after it was appointed.
It also told the French group that TIM was executing a plan that was "devised and approved by Vivendi" and urged the French group to work towards constructive solutions at the board level.
Under Genish, who was appointed when Vivendi controlled the board, TIM is working on a three-year turnaround plan focussing on a digital transformation and fixing the group's finances.
However, the former state phone monopoly has been facing growing challenges in both fixed and mobile: Broadband group Open Fiber is rolling out a rival fibre optic network while French telecoms group Iliad in May launched its low-price mobile offer for Italy.
TIM also committed this week to spending more than 2.4 billion euros ($2.77 billion) - much more than initially expected - on fifth-generation mobile spectrum in Italy, in a move that is likely to raise its debt pile and could affect its ambition to return to an investment grade credit rating next year.
(Reporting by Agnieszka Flak and Sudip Kar-Gupta, Editing by William Maclean)