Mancini, one of the highest-profile foreign coaches to join the Russian Premier League in recent years, signed a three-year contract with Zenit in 2017.
However, the club said the contract, which will come to an end on Monday, had been severed through mutual consent and without any compensation payable.
This month, Italian soccer federation commissioner Roberto Fabbricini said Mancini had held talks with Italian soccer officials but that no deal had been reached.
"I want to emphasise that the move to leave Zenit was exclusively my own," Mancini told the club's website.
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"Much of it has to do with family circumstances. The decision to leave was made some three months ago. It has nothing to do with the club or the national team."
Mancini added that he currently had no agreement with the Italian team and could not confirm he would be named to the post.
"It would have been wrong to negotiate before the contract was broken, which was announced today," he said. "It would be against the rules."
Zenit, bankrolled by Russian energy giant Gazprom, have used their deep pockets to attract foreign managers in the past, including Dick Advocaat, Luciano Spalletti and Andre Villas-Boas.
Mancini, who has coached a number of elite sides including Inter Milan, Galatasaray and Manchester City, led the latter to a first English title in 44 years in 2012. Zenit are currently fifth in the Russian Premier League with one game remaining.
Italy have a managerial vacancy after Gian Piero Ventura was sacked when the four-times champions failed to qualify for this year's World Cup finals taking place in Russia from June 14 to July 15.
(Reporting by Gabrielle Tétrault-Farber; Editing by John O'Brien and Pritha Sarkar)
