Former Lazio boss and Italy international Inzaghi, 45, has signed a two-year deal at the San Siro.
Tottenham are believed to have turned their attention to Conte, who left Inter after leading them to the Serie A title this season.
Premier League: Tottenham Hotspur hold talks with former boss Mauricio Pochettino over return as manager, say reports
The 49-year-old left Spurs in November 2019 after five years in charge and was replaced by Jose Mourinho, who was sacked last month.
Serie A: Antonio Conte parts ways with Inter Milan by 'mutual consent' after guiding club to first league title in 11 years
The 51-year-old Italian has been at loggerheads with the cash-strapped club's owners Suning over planned cost cutting which would prevent him building the team he wants to challenge at home and in Europe.
Conte still has a year left in his contract, but has been at loggerheads with the cash-strapped club's Chinese owners Suning over planned cost cutting
A look at the winners and losers of the 2020-2021 Serie A campaign which ended on Sunday.
European football matchday: Chelsea meet Leicester City in FA Cup final; Juventus face Inter Milan in bid for top-four
Here's a look at what's happening in European football on Saturday.
Serie A: Five-goal Inter Milan celebrate title with record 14th consecutive home win; Napoli move second
Since losing against Sampdoria, Inter Milan have gone 18 games without defeat and have now won 14 games in a row at home for the first time.
It is a fifth league title for Conte who won three with Juventus between 2012 and 2014 and one with Chelsea in 2017.
Antonio Conte's side clinched their 19th 'Scudetto' after nearest rivals Atalanta drew 1-1 at Sassuolo to leave the 'Nerazzurri' 13 points clear with four games remaining.
Antonio Conte's Inter are 13 points ahead of city rivals AC Milan who reclaimed second place with a 2-0 win over 18th-placed Benevento, with four games to play.
Serie A: Matteo Darmian pushes Inter Milan to brink of A title with 'nine point' win, Atalanta move second
Inter are 11 points ahead of Atalanta, who moved second with a 5-0 rout of 10-man Bologna, with five games remaining.
"Meritocracy must come first. Having said that, after everything that has happened, UEFA, they have to reflect too," said Conte.
Serie A: Matteo Darmian's strike helps leaders Inter Milan beat Cagliari; Juventus clinch win against Genoa
Moroccan Achraf Hakimi came off the bench and set up Darmian to grab the winner with 13 minutes to go at the San Siro.
Coach Antonio Conte ran onto the pitch to celebrate with his players as the goal restored Inter's 11-point advantage over second-place AC Milan, which won 3-1 at Parma on Saturday.
Romelu Lukaku had a role in both the goals by his former Manchester United teammate to give Antonio Conte's side a sixth consecutive league win.
Serie A: Romelu Lukaku scores inside first minute as Inter Milan pull seven points clear; AC Milan beat Roma
Atalanta shook off their Champions League defeat to Real Madrid midweek with a 2-0 win at Sampdoria to move up to fourth.
Inter opened up a four-point lead over their second-placed opponents as Antonio Conte's side target a first league title since their unprecedented treble under Jose Mourinho in 2010.
Gianluigi Buffon, who escaped a ban for the offence, was overheard using a "blasphemous expression" towards teammate Manolo Portanova during Juve's 4-0 league win over Parma
Coppa Italia: Inter Milan coach Antonio Conte apologises for 'reacting badly to provocation' in Juventus defeat
Ex-Juve boss Conte appeared to show the finger to his former club's directors at half-time, while footage from broadcaster RAI at the end of the match showed the hosts' chairman Andrea Agnelli apparently telling Conte to "shut up".