Macquarie Infrastructure and Real Assets' (MIRA) offer, which is targeting a stake of between 35% and 50%, will be sent to Enel before the utility's board meeting set for Thursday, the source said.
But the source added Enel would not necessarily discuss the offer at the meeting on Thursday.
Macquarie and Enel declined to comment.
The Italian government is trying to broker a deal between former phone monopoly Telecom Italia (TIM) and smaller rival Open Fiber to merge their fibre assets and create a national company.
In June Enel said its board had received a non-binding offer from MIRA to buy the whole or part of its stake in a bid that sources said valued the company at 7.7 billion euros (7.02 billion pounds).
On Tuesday Italian newspaper Il Messaggero said the fund's investment committee was set to sign off on a bid that valued Open Fiber at more than 8 billion euros including debt.
State lender Cassa Depositi e Prestiti (CDP), which owns the other 50% of Open Fiber, has a right of first refusal on any sale by Enel.
Sources have previously told Reuters Rome wants Enel to make sure that CDP ends up with a majority stake in Open Fiber.
In case of an ownership change, Vodafone and WindTre also have a right of first refusal on a combined stake of around 15%.
"There are a lot of moving parts and the bid will be structured to take this into account," the source said.
Italy's ruling coalition has placed the creation of a single full-fibre network to close Italy's digital divide with other European countries at the heart of its policies.
Last month it set out a road map to try and overcome deadlock between TIM and Open Fiber and create a high-speed broadband network open equally to all operators and, crucially, acceptable to regulators.
By Stephen Jewkes