The original deadline for a binding bid was the end of September, but that is set to be moved back to October 15 at KKR's request.
The potential deal is explained by pointing out the value of the deal.
TIM is offering NetCo, a venture that includes both TIM's domestic fixed-access network and international wholesale telecoms operator Sparkle.
TIM's network covers almost 89 percent of the nation's households, and its fibre cable spans over 23 million kilometres across the country.
Sparkle's cable network spans over 600,000 km and it has a direct presence in 32 countries.
With 26 billion euros in net debt and cash burn, TIM has resorted to Hiving off its main asset, in a move that would be a major European country among former phone monopolies.
The sale is the main plank in Chief Executive Pietro Labriola's plan to reshape the company, whose earnings have been slashed for years due to intense competition on its home ground. In many cases, such initiatives have failed to produce results.
TIM's network is Italy's main telecommunications infrastructure, and governments have been searching for years to ensure investments are carried out to upgrade it to fast fibre optics from copper.
Vivendi, the largest individual investor in France, has a net worth of almost 24 percent, making it the largest French media group. The company is trying to argue that KKR's bid undervalues the network and would put at risk the sustainability of the remaining services business.
Vivendi, which IPO TIM in 2015, has been repeatedly asked to write down the value of its holding and faces a 75 percent theoretical loss on its initial 4 billion euro investment.
State lender CDP, with a stake of almost 10 percent, is the second-largest investor.
The government of Prime Minister Giorgia Meloni is also involved, as it views telecoms infrastructure as a crucial national asset. The Treasury plans to take a 15-30 percent interest in the NetCo business being sold.
Trade unions are also worried about safeguarding the jobs of more than 40,000 TIM workers in Italy.
Beyond TIM's grid, Open Fiber is being rolled out by CDP and Australian investment group Macquarie, a company controlled by CDP and Australian investment group Macquarie.
There have been long talks of combining TIM's network with Open Fiber, but competition concerns have hampered such a deal so far.
The EU's annual report on digitalisation said that the share of households having access to full fibre-optic connectivity in Italy is 44 per cent, well below the EU average of 70 per cent.
When a binding bid is submitted, getting Vivendi's support will be crucial for ensuring a smooth completion of the sale.
With its 24 per cent vote, Vivendi could throw a spanner in the works at any TIM shareholder meeting to vote on a deal or challenge it in the courts.