Conte agreed to the mutual termination of his contract on Wednesday, and Sky Italia reported on Thursday that Inzaghi has agreed a two-year deal to coach Inter when his stint in Rome comes to an end next month.
"With great emotion I want to inform you that on June 30 my wonderful adventure with Lazio will come to an end," Inzaghi told ANSA.
"I thank the club, the president, the players and the fans who have accompanied me over these splendid 22 years as a player and coach. We fought and won together.
"These colours will forever remain in my heart: blue and white will always be part of my soul."
Inzaghi has been at Lazio since joining as a player in 1999, winning the Serie A title and three Coppa Italias as a striker before hanging up his boots in 2010.
He started coaching in the capital club's youth sector before taking the first-team job in April, 2016.
Inzaghi guided Lazio to the Coppa Italia and two Italian Super Cup successes, as well as leading them to the last 16 in their first Champions League campaign in 13 years this season.
Sky Italia reported that Inzaghi reached a verbal agreement to extend his Lazio contract during a meeting with club president Claudio Lotito on Wednesday, but was later convinced by Inter to change his mind.
"We respect the reconsideration of a coach and, before, a player who for many years has tied his name to the Lazio family and to many Biancocelesti successes," Lazio said in a statement.
Inter will confirm on Friday that Inzaghi has signed a two-year deal worth €4 million a season, Sky Italia reported.