TIM and Open Fiber, a wholesale-only broadband unit jointly owned by government-controlled utility Enel and state lender CDP, have been talking since June last year on ways of combining their assets, but talks have so far been fruitless.
In a bid to break the stalemate, Italian Economy Minister Roberto Gualtieri last month asked parties involved to agree on a memorandum of understanding (MOU) by the end of July, but sources said more time was needed to strike a deal.
Under the planned deal with KKR, TIM is set to sell the fund a minority stake in Fibercop, a vehicle that will hold the incumbent's secondary network that is made up mainly of copper.
TIM said in a statement that KKR's binding offer, which the board of the former phone monopolist discussed on Tuesday, would buy 37.5% of the unit for 1.8 billion euros (1.62 billion pounds), while Swisscom's Fastweb would hold a 4.5% stake.
TIM, whose main shareholder is France's Vivendi, said its board had mandated Chief Executive Luigi Gubitosi to negotiate with the government steps to speed up the creation of a single network and would decide on KKR's binding bid on Aug.31.
TIM added that the group's last-mile project deal could change in the light of the negotiations with the Italian authorities.
KKR agreed to extend its binding offer for TIM's secondary grid until end-August at the request of the Italian government, two sources close to the matter said.
KKR declined to comment.
The prime minister's office said Rome had no objection to KKR, but added it was in favour of including other institutional and market players in a broader deal aimed at creating a single national network.
Sources told Reuters last month that CDP, which also own a 10% stake in TIM, could invest in TIM's last-mile network to allay Rome's concerns over the planned sale of part of the grid to KKR.
Reuters wrote on July 27 that Rome drafted a plan under which any combined merger of TIM's assets with Open Fiber could initially be majority owned by Telecom Italia but would grant equal access to all market players.
TIM said its organic core earnings in the second quarter fell 6.4% from a year earlier to 1.761 billion euros, broadly in line with a company provided consensus of 1.735 billion euros.
($1 = 0.8475 euros)
By Giuseppe Fonte and Elvira Pollina