A series of individual mistakes have cost Spurs in recent weeks and there are serious doubts about the Italian’s long-term future.
Conte’s contract expires at the end of the season and Standard Sport understands the former Chelsea manager wants a significant pay-rise to extend his stay.
Currently five points off the top four places having a played game more than Manchester United and Newcastle, missing out on the Champions League would surely also complicate the future of Harry Kane.
Still, even with such problems, Carragher believes Spurs have far more pressing needs.
Speaking after Thursday’s defeat at the Etihad Stadium, the Sky Sports pundit said: “He’s got those great attacking players, but his goalkeeper and defenders are nowhere near good enough for a Conte team to be successful, and that’s where they’ve got major problems.
“The goalkeeper at Tottenham right now, he’s been a good goalkeeper, but he has become a liability, they need a new goalkeeper for next season.
“Antonio Conte is talking about taking them further, getting top four or challenging for big honours.
“They need a new goalkeeper, need two centre-backs and a right wing-back and that’s not to come into the squad, that’s to play.”