According to Bloomberg, who quote sources close to the matter, the two firms have held exploratory talks about the offer and have said the new bid aims to share the financial load of deal.
The news comes days after KKR launched a €10.8 billion ($12 billion) takeover bid for TIM in attempt to take the company private. Since the news broke, shares in the company have soared reaching 32% higher at €0.4586, valuing shares at €0.505 each, a premium of approximately 45.7% based on the company's closing price on Friday 19 November.
As of Friday 26th November, TIM shares rose by 2.8% and they were up 0.9% to €048.82 by 10:38 am in Milan.
At the time of the offer, it was rumoured that CVC Capital and Advent International were open to solutions to strengthen the company, with a spokesperson for both companies saying the two "are open to dialogue with all shareholders to identify a transparent solution" to strengthen TIM.
The deal has also been met from strong opposition from TIM's largest shareholder Vivendi, who following two profit warnings over the last three months has been keen to oust TIM CEO Luigi Gubitosi. Vivendi has also indicated that the initial KKR offer was too low.
According to ANSA news agency, Gubitosi has since said he is prepared to step aside in order to allow the company to properly assess KKR's takeover proposal – essentially tendering a resignation.
In a letter to the company's board, Gubitosi advised that the board shouldn't delay its decision making which could be seen as defending interests of certain shareholders.