The Toledo Museum of Art (TMA) in the US state of Ohio said it will return the artworks, including a bronze sculpture and carved stone.
TMA will coordinate the return of four objects with US representatives of the Republic of India, the museum said in a statement.
"We look forward to working with the Indian government to return these objects to their rightful home," Director Brian Kennedy said.
For 35 years, Kapoor owned and operated the Art of the Past gallery in Manhattan, which sold Asian antiquities to many museums and private collectors around the world.
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Kapoor was arrested in Germany in October 2011 and extradited in July 2012 to India, where he is incarcerated awaiting trial.
The four objects to be returned to India are: a stone stele of Varaha Rescuing the Earth (acquired in 2001); a previously announced return of a nearly 1,000-year-old bronze sculpture of the Hindu god Ganesha described as being from Tamil Nadu (acquired in 2006).