Saoirse Kennedy Hill, 22, was at the compound, the home of her grandmother, Ethel Kennedy, 91, on Thursday afternoon, when emergency personnel were called.
She was taken to Cape Cod Hospital in Hyannis, after what is believed to be an overdose, The New York Times said, citing family and friends.
According to the newspaper, she was pronounced dead at the hospital. In a statement released to media, the Kennedy family confirmed the young woman’s death.
"Our hearts are shattered by the loss of our beloved Saoirse," the family statement said. "Her life was filled with hope, promise and love."
The statement quoted her grandmother as saying: "The world is a little less beautiful today."
The Cape and Islands District Attorney's Office confirmed police officers had been called to the residence and the matter remained under investigation.
Ms Hill was a student at Boston College, where she was a communication major and vice president of the Student Democrats, according to The New York Times.
It said she had written about struggling with mental illness while a student at a private preparatory school in Massachusetts, the Deerfield Academy, in 2016.
Ms Hill was the only child of Paul Michael Hill and Courtney Kennedy Hill, who was the fifth of 11 children of Ethel and Robert Kennedy.
Her death is the latest in a long series of tragedies that has beset one of America’s political dynasties.
In 1968 Robert F. Kennedy, then a US senator for New York, was assassinated in Los Angeles just after winning the California primary race for the Democratic presidential nomination.
Two of Robert and Ethel Kennedy’s sons suffered untimely deaths – David Kennedy, of a drug overdose in 1984 at age 28 and Michael Kennedy in 1997 at age 39.
RFK's older brother, President John F. Kennedy, was himself assassinated in 1963, in Dallas, and his son, John Jr., Kennedy died with his wife and sister-in-law when the plane he was piloting crashed in the Atlantic off Martha's Vineyard, not far from Hyannis Port in 1999.
Ms Hill's death comes weeks after the 50th anniversary of the scandal surrounding a fatal car accident in which RFK's younger brother, then-Senator Edward Kennedy, drove off a bridge connecting the tiny island of Chappaquiddick to Martha's Vineyard, killing his passenger, Mary Jo Kopechne.
The murky circumstances of that crash effectively dashed Edward Kennedy's presidential ambitions.
Additional reporting by Reuters