Police were interviewing her neighbours, a police officer said, adding that no suicide note was found.
"We suspect she hung herself at her home" late on Monday, the officer told AFP, saying her body has been sent for a post-mortem examination.
Khan, who was born in New York before moving to London and then Mumbai, made her Bollywood debut in 2007 with the controversial film "Nishabd" (No words), playing the role of a teenager in love with her best friend's much older father, played by Bachchan.
"It was quite a controversial subject in a country which is still pretty closed on aspects of sex and relationships," Khan later recalled in an interview.
The movie won attention because of its provocative storyline and Khan was praised for her bold acting.
Khan, who changed her name from Nafisa Khan, went on to star alongside actor Aamir Khan in "Ghajini", a box-office hit.
India's film industry reacted with shock and sadness to her death.
Actor Sonam Kapoor tweeted: "Nobody should be in so much pain and a state of such hopelessness. I hope her soul finds some respite in death. #ripjiahkhan."
Actor Aamir Khan said he was "deeply pained to hear this very tragic news. Jiah is a close family friend and I can only imagine what grief her mother and family must be going through".
"I wish I had known that she was troubled and needed help," he told AFP.