The Italian boss eased doubts over his Chelsea future by putting pen to paper on a new two-year contract worth £19.2million until 2019.
However, despite Conte receiving a salary increase he did not sign a contract extension.
And according to Yahoo Sport, the 48-year-old came close to leaving the Blues after his first season in charge that saw him lead the club to Premier League glory.
A source says that at a certain point “there seemed no doubt he was going to leave Chelsea” and those close to him weren’t expecting Conte to commit his future.
Things haven’t improved at Chelsea since he signed his new deal with Conte publicly voicing his concerns over a lack of new signings this summer.
The west London outfit remain on the lookout for fresh talent with the window closing on August 31 but the season has already started and Conte is frustrated at the options he has been left with.
Conte’s concerns were proven true when they lost their Premier League opener 3-2 to Burnley at home on Saturday.
And it seems Conte and the board’s relationship could deteriorate further if results don’t pick up and players aren’t signed in the next fortnight.
“This is the kind of situation Antonio cannot cope with,” a source told Yahoo Sport. “When he starts losing matches that is when he gets really crazy.
“It’s impossible to assess what it will take to calm him down.
“When Antonio is in such a state of mind he makes himself hard to cope with everyone, and everything.
“He starts getting angry at staff, players, and others.”
Chelsea will look to get their first win of the season under their belts when they take on Tottenham at Wembley on Sunday (4pm)