The 19-year-old, on loan from Tottenham Hotspur, damaged his ankle in the midweek Capital One Cup win against Doncaster Rovers.
Canaries boss Hughton admitted that the news is a real blow to his attacking options.
"Harry is being assessed. He will certainly be out for a period of weeks, a month-plus rather than short of that," he said.
"It is a real disappointment, especially when he was just up and going. He has settled in very well. We just hope it is a shorter period out rather than a longer one."
Kane made his debut for Norwich as an impact substitute in the league match against West Ham United at Carrow Road a fortnight ago.