The 86-year-old had been ill for a short time.
Forman was also known for directing Hair, Ragtime and The People vs. Larry Flynt.
He received numerous honours, including a Directors Guild of America's Lifetime Achievement Award in 2013 for motion picture direction.
Upon hearing about the news, Elton wrote on his Instagram page:
''So sad to wake up this morning to hear of Milos Forman’s passing. He was an extraordinary film maker and a dear friend. His films were bold and brave cinematic explorations of the human condition. We came very close to collaborating. It would have been a wonderful experience. My heart goes out to his beautiful wife Martina and all of his children.''