"According to multiple sources ... [they] are all but officially finished," the magazine said.
"Johnny has started reaching out to lawyers, probably to quietly discuss how to get out of the relationship," another source told website RadarOnline.
"They're not married but they've been together for years and have kids together so it isn't as easy as just breaking up.
"Johnny is so talented at acting, but he doesn't seem able to hide how badly things are going right now."
Reps for either party have yet to comment on the reports.
Speaking after the red carpet premiere of her flick this week, Paradis refused to directly address tabloid speculation that she and Depp are calling it quits.
"You know, when I eat three peas, I'm pregnant. When I visit a city, I'm buying a house," she told radio station Europe 1 when probed about the rumoured split.
"In the winter I separate, in the summer I marry. It's been 15 years since I've been getting married every year. In addition I have to answer all these rumours!"
Touché!
In other words, MYOB.
Depp is believed to be in Los Angeles with the couple's two children, Lily-Rose, 12, and Jack, 9.
Earlier in the week, the Joe Le Taxi singer talked bout her new film and revealed she did not believe in marriage or soul mates because "if you lose your soul mate everything is done for".
Paradis, who plays a single mother in the movie, added: "Café de Flore speaks of love, its joys, its pains and its dramas - to love and to lose. This story upset me, I was upside-down, in the depths of myself.
"The idea of a soulmate is beautiful and very romantic to talk about it in a movie or a song, but in reality, I find it scary."
And so the plot thickens ...
Check out the trailer for Café de Flore (Warning: contains swearing):
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