Another 15 American museums have been identified as holding items obtained from Mr. Kapoor, but many said in the interviews this week that they had researched their Kapoor holdings and were satisfied that their items were not stolen or that they wanted to see proof of illegality before returning the 500 or so objects in question. Among those museums are the Los Angeles County Museum of Art, the Asian Art Museum in San Francisco, the Metropolitan Museum of Art and the Art Institute of Chicago.

“The ones that could be suspect were looked into and at the moment we are convinced that they are O.K.,” said Miranda Carroll, spokeswoman for the Los Angeles museum, which has 62 objects connected to Mr. Kapoor.

Tim Hallman, spokesman for the San Francisco museum, which has four Kapoor items, said, “We don’t believe any of the objects are a particular problem and no official claims have been made on any of them.”

The federal investigation, Operation Hidden Idol, began in 2012 with raids on Mr. Kapoor’s gallery and on several warehouses and other locations where he stored Indian antiquities. Officials with Immigration and Customs Enforcement and Homeland Security Investigations, which are jointly directing the investigation, said they had seized tens of millions of dollars worth of objects that they believed were looted from ancient temples in India.

In addition, the Manhattan district attorney’s office, working with the investigators, has prosecuted Mr. Kapoor’s sister; a woman identified as his girlfriend; and the longtime manager of his gallery on charges related to the artifacts. The manager, Aaron Freedman, pleaded guilty in December 2013 to six criminal charges and is said to be cooperating with the authorities.