TIM's board unanimously mandated CEO Pietro Labriola to start talks with KKR aimed at obtaining a binding offer no later than Sept. 30, the group said in a statement.
KKR, whose bid could be worth around 23 billion euros ($25.19 billion) in total, prevailed over a consortium formed by Italian state lender Cassa Depositi e Prestiti (CDP) and Australian fund Macquarie.
A number of efforts have been made over the years to restructure TIM, Italy's former national phone monopoly.
The U.S. fund has already invested 1.8 billion euros in the grid and had a bid to take the entire company private rejected last year.
It is now seeking to buy a unit comprising TIM's entire domestic fixed access network and submarine cable business Sparkle.
A deal to sell the grid will need the approval of the Italian government, which can use a "golden powers" rule to set conditions or block bids for strategic assets such as TIM's network.
To win Rome's nod, KKR is open to Italy's Treasury or other state-backed entities such as infrastructure fund F2i becoming a co-shareholder in the network company, people familiar with the matter have previously said.
The network sale is a key plank of Chief Executive Pietro Labriola's strategy to relaunch TIM by slashing the former phone monopoly's 25 billion euro debt.
But Labriola's plans have run into strong resistance from Telecom Italia's top investor Vivendi, the French media group, which wants the landline grid - the company's main asset - to fetch at least 31 billion euros in any sale.
With its 24% stake in TIM, Vivendi also wants any decision on the grid to go through an extraordinary shareholder vote, which would make it easier for it block a deal.
The exact value of KKR's bid hinges on the terms of the contracts between the grid it will take over and TIM's remaining business as well as some performance targets.
($1 = 0.9131 euros)
(Reporting by Elvira Pollina; writing by Francesca Landini, editing by Alvise Armellini and Deepa Babington)
By Elvira Pollina
