Argento, 47, posted a picture wearing a T-shirt featuring professional bodybuilder Ronnie Coleman with the words "Stop Busting My Balls" superimposed over the image. She sent the same message to her late partner the night before he died, according to the book Down and Out in Paradise by Charles Leerhsen (out Oct. 11).
Bourdain died by suicide in June 2018 at age 61 while in Kaysersberg, France.
"Is there anything I can do?" Bourdain asked Argento in the conversation, which is included on the first page of the forthcoming biography.
"Stop busting my balls," Argento wrote back, to which Bourdain then replied: "Okay."
The next day, Bourdain was found dead in his hotel room, where he was on location filming an episode of his series Parts Unknown with close friend and French chef, Eric Ripert.
The book features intimate details — taken largely from texts and emails from Bourdain's phone and laptop — and charts Bourdain's rise to fame, from a Vassar College dropout to a food superstar.
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During his final days filming his CNN series with Ripert, Bourdain fought with the Italian actress while he was on set, and had to leave the filming "multiple times to talk to her on the phone," according to the book, which PEOPLE excerpted exclusively on Wednesday. Bourdain became angry after she was spotted by paparazzi with French journalist Hugo Clément.
Matthew Eisman/Getty Anthony Bourdain and Asia Argento
"Things escalated on Wednesday when by all accounts she told him she no longer wanted to be with him. Everyone was keeping an eye on him all day and night because he was incredibly distraught. More screaming phone calls through the day. By Thursday he seemed to be better and kind of wanted everyone to back off," reads the excerpt.
Argento did not respond to PEOPLE's request for comment, but told the New York Times, "I wrote clearly to [Leerhsen] that he could not publish anything I said to him."
Three months after Bourdain's death in 2018, the actress spoke to DailyMailTV after she faced backlash following Bourdain's death.
"People say I murdered him. They say I killed him," she said through tears in the interview. "People need to think that he killed himself for something like this? He had cheated on me too. It wasn't a problem for us. He was a man who traveled 265 days a year. We took great pleasure in each other's company when we saw each other. But we are not children. We are grown-ups."
If you or someone you know is considering suicide, please contact the 988 Suicide and Crisis Lifeline by dialing 988, text "STRENGTH" to the Crisis Text Line at 741741 or go to 988lifeline.org.