The deal, expected in early April, the Reuter's sources said, would help Eni reassure investors that it has enough cash to continue paying its dividend and advance the Kremlin's strategy of bringing key resources back under state control.
Eni's shares were up 3.6 percent as investors had feared that Gazprom could fail to meet or would ask to extend the deadline for the buyback, citing a lack of refinancing options and the fact that Gazprom Neft's value has tumbled since it took the call option.
Gazprom Neft's shares closed up 14 percent on the MICEX after rising as high as 30 percent in early trade on hopes that Gazprom would offer a generous buyout to minority shareholders of Gazprom Neft.
The deal would mark the first major refinancing help provided by Russian state banks since the Kremlin abruptly suspended direct industry bailouts in February.
Eni could become the outright owner of the assets if Gazprom misses the deadline.
Gazprom must either exercise the buyback before April 9 or agree on an extension to the deadline.
It can buy 20 percent of Gazprom Neft from Eni and 51 percent in a joint venture operating Siberian gas assets that belongs to Eni and its partner, Enel.
The Reuter's sources said, however, that the two sides had yet to reach an agreement on the final price. Gazprom and Eni declined official comment.
Eni and its partner at the auction, Italian utility Enel, paid a total of $5.8 billion for the assets and agreed to resell the Gazprom Neft stake for around $4 billion.
One-fifth of Gazprom Neft is worth about $2.2 billion at current market prices.