Conte is considered a legend by the 'Bianconeri' part of the northern Italian city having won three consecutive league titles as coach of Juventus.
But despite being met with warm applause upon his return to Turin on Sunday, the 45-year-old is showing signs of discontent only months after replacing Cesare Prandelli following Italy's disastrous World Cup in Brazil.
Italy could only eke out a 2-2 draw against Bulgaria on Saturday, a result which left them second in Group H qualifying for Euro 2016, two points adrift of leaders Croatia.
However, their performance at the Vasil Levski stadium left Conte with perhaps more questions than answers.
After trailing through an own goal in the fourth minute, Bulgaria struck twice in the space of six first-half minutes following two incisive counter-attacks that exposed holes in Italy's defence.
Conte has since brought in reinforcements, with AC Milan defender Ignazio Abate and Inter Milan defender Davide Santon called to plug the gaps, although if used they could be handed wingback roles in Conte's preferred 3-5-2.
"We conceded two goals after two counter-attacks, and that's something we have to improve," said Conte, who could welcome number one goalkeeper Gianluigi Buffon back to the fray after he pulled out late on Saturday with a fever.
On the pitch, Conte's biggest dilemma appears to be his attack.
Sassuolo's Simone Zaza and Ciro Immobile of Borussia Dortmund failed to hit the target in Sofia, where a late leveller by Brazil-born Sampdoria forward Eder was required to secure a share of the points for the visitors.
Conte could be tempted to reshuffle his starting line-up for Tuesday, with Eder, Manolo Gabbiadini, Graziano Pelle and Franco Vazquez all waiting in the wings.
Off the pitch, the jury is also still out on Conte who has fought Italy's Serie A clubs to have more regular access to his squad, although his efforts so far have been in vain.
Reports on Sunday claimed Conte had also been refused permission by Italian federation (FIGC) president Carlo Tavecchio to commit to his role part-time, to allow him to also coach a club side.
Conte last week was also a target for fans unhappy at his decision to call up Eder and Argentinian-born Vazquez to his squad at the expense of Italy-born players.
Juventus defender Giorgio Chiellini said it was time the coach was given a break.
"Conte and the Italy set-up don't deserve to be in this situation," he said.
"Everyone has to look at the national team in a different light, without always being critical, to allow it to grow."
Conte said he would remember the flak aimed his way for a long time.
"A lot of what has been written hurt me, but I have a memory like an elephant," he said.
"I've worked hard throughout my life, and that's what I intend to keep on doing with Italy, if I am left to do my job in peace."
England come to Italy on the back of a one-sided 4-0 win over Euro 2016 group rivals Lithuania, in which Tottenham sensation Harry Kane scored only 80 seconds into his debut.
However, Roy Hodgson's side will also be looking to avoid a third consecutive defeat to the Azzurri, having been knocked out of Euro 2012 by Italy and suffered a 2-1 defeat in the group stages of last year's World Cup.
Danny Welbeck is sidelined with a knee injury and Daniel Sturridge has succumbed to a hip problem, so Kane could follow up his England debut by starting alongside England's only other fit striker, Wayne Rooney.