Peaches Geldof (file pic)
The model died suddenly on April 7, aged 25, but the autopsy proved inconclusive.
However, Detective Chief Inspector Paul Fotheringham is due to tell a hearing in Gravesend, Kent, later Thursday that toxicology tests showed she had died of a heroin overdose, the paper reported.
Peaches, who wrote for British magazines and newspapers and presented celebrity-driven television shows, was found dead at her countryside home where she was looking after 11-month-old son Phaedra.
Husband Thomas Cohen, 23, had spent the previous night apart from his wife and was looking after their eldest son, 23-month-old Astala. He raised the alarm when he could not contact her by phone.
Police searched the property for drug paraphernalia, but found none.
Supermodel Kate Moss was among a host of celebrities who attended last month's funeral at an English village church filled with memories for the tragedy-hit family.
Geldof's body was carried into St Mary Magdalene and St Lawrence church in Davington in Kent, southern England, in a white coffin painted with a blue sky and clouds.
The church, which is next to the Geldof family estate, was the venue for the funeral of Peaches' mother Paula Yates, who died in similar circumstances in 2000.
The TV presenter accidentally overdosed on heroin while looking after four-year-old daughter Heavenly Hiraani Tiger Lily at their London home.
Peaches posted online an old photograph of her with her mother the night before her death.
Bob Geldof has said his family is suffering "beyond pain" at the death of his second daughter.
A self-confessed "wild child" in her youth, she had become an advocate of "attachment parenting" after becoming a mother.
In her final column for Mother and Baby magazine, published on Tuesday with the Geldof family's blessing, Peaches wrote that she was "happier than ever".
She said she used to live a life of "wanton wanderlust... lost in a haze of youth and no responsibilities", with "nothing stopping me from having constant fun".
But that life became "boring" and her two children "became her entire existence and saved me" from a life of "pure apathy".
She said fair-weather friends drifted away, but in a "magic moment" when one of her children brought her a picture he had made, "all my doubts were erased".
"I had the perfect life - two beautiful babies who loved me more than anything. It was, and is, bliss," she wrote.
"I'm happier than ever. Right now life is good. And being a mum is the best part of it."
Coroner Roger Hatch is set to adjourn the inquest until July in order to give relatives a chance to attend the full hearing, the Times reported.
RESULT
Peaches Geldof, daughter of Bob Geldof, as she arrived for the European premiere of The Twilight Saga: Breaking Dawn Part 2 in London on Nov. 14, 2012 Luke MacGregor
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