Birkin had been suffering from health problems in recent years that had forced her to cancel concerts.
The cause of her death was not immediately known. According to a police source, a nursing assistant found Birkin dead at her Paris home on Sunday.
With no suspicious elements found on her body or in the apartment, no investigation was to be opened, the source added.
Her agent and family were yet to confirm the death.
Birkin was catapulted to fame through her turbulent relationship with legendary singer-songwriter Serge Gainsbourg and her heavily accented French, which became one of her signatures in her adopted home.
With her flared jeans, mini dresses and distinctive fringe, Birkin was the ultimate It girl in the 1970s.
Once photographed in a dress that appeared transparent under the camera flashes, she quipped: "If I had known I wouldn't have worn underwear."
In 1984, Hermes named one of its handbags after her, after she complained to a former chief executive that she could not find a leather bag that could fit all she needed as a young mother.
The Birkin became an eye-wateringly expensive and sought-after status symbol.
The luxury brand in a statement hailed Birkin as "a dear and intimate friend" who was endowed with a "great humanity".
French President Emmanuel Macron called Birkin "a complete artist" who "sang the most beautiful words in our language".
"Unimaginable to live in a world without your light in it," singer Etienne Daho, one of her close friends, wrote on Instagram.
French Prime Minister Elisabeth Borne hailed her as an "unforgettable icon, a unique voice and charm" who "with her music and her talent transcended generations".
Hailing an artist "eternal in our hearts", actor and activist Brigitte Bardot said "when someone is so pretty, so fresh, so spontaneous, with the voice of a child, they don't have the right to die."
"There was something about her that touched the French spirit," artist and embroiderer Jenny Hunt said outside the house Birkin once shared with Gainsbourg in Paris.
"It's very hard to articulate, but you feel it when you see it."
Born in London, Birkin crossed the Channel in 1968 at the age of 22 to star in a film alongside Gainsbourg, who was 18 years her senior.
It was the start of a 13-year relationship that made them France's most famous couple, in the spotlight as much for their bohemian and hedonistic lifestyle as for their work.
Birkin, with her soft voice and androgynous silhouette, recording a steamy duo with a growling Gainsbourg in 1969, "Je t'aime... moi non plus".
Banned on radio in several countries and condemned by the Vatican, the song was a worldwide success.
"He and I became the most famous of couples in that strange way because of 'Je t'aime' and because we stuck together for 13 years and he went on being my friend until the day he died. Who could ask for more?" Birkin told CNN in 2006.
"Paris became my home. I've been adopted here. They like my accent," she said.
The daughter of an actor and a naval officer, Birkin first married James Bond composer John Barry, with whom she had a daughter, Kate. The marriage lasted three years.
She moved to France after meeting Gainsbourg on the set of a romantic comedy in which he was her co-star.
They had a daughter, Charlotte, who became a hugely successful actor and singer.
The musical and romantic relationship between Gainsbourg and Birkin was famously tempestuous.
During one of their raging rows, Birkin tossed herself into the River Seine after throwing a custard pie in Gainsbourg's face.
She blazed her own trail after walking out on France's favourite bad boy in 1980.
In her around 70 films she worked with leading directors including Bertrand Tavernier, Jean-Luc Godard, Alain Resnais, James Ivory and Agnes Varda.
But she remained forever associated with Gainsbourg, who continued to write songs for her after their split, including "Les dessous chics" about lingerie being used to try to cover up a relationship on the rocks.
"It's the most beautiful song about separation you could ever have," a statement by Birkin in a 2018 interview.
Her life was marked by tragedy and obstacles.
Her oldest daughter, Kate Barry, a photographer, died in an apparent suicide in 2013.
She fought leukaemia in the late 1990s and in 2021 suffered a minor stroke.
Birkin is survived by her daughter with Gainsbourg, Charlotte, and another daughter, Lou Doillon, a singer, from a 13-year relationship with French director Jacques Doillon.
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