The singer, who shares son Angelo with her ex-husband Simon Konecki, confirmed her relationship with Paul in September 2021.
She stars on the cover of the October 2022 issue of Elle UK and its American counterpart, Elle US, with a cover image showing her posing in a white coat, as well as featuring on global Elle covers across the world.
She said of Paul: “I’ve never been in love like this. I’m obsessed with him.”
Forty-year-old Paul is the founder of management company Klutch Sports Group, which represents high-profile basketball stars including LeBron James, Anthony Davis and John Wall.
Adele, who headlined Glastonbury festival in 2016, announced her split from charity boss Konecki in April 2019.
Their divorce was finalised in March 2021, with the former couple sharing custody of their son.
The 34-year-old pop superstar, who now lives in Los Angeles, said she does want to get married again but confirmed she is not engaged to Paul.
She added that while she does want to have more children, her focus at the moment is on her upcoming Las Vegas residency.
In July she announced the new dates for the delayed Las Vegas gigs, saying they would take place from November to March.
She had been due to perform a series of shows from January to April this year at Caesars Palace but they were scrapped at short notice.
She said: “I definitely want more kids. I’m a homemaker and I’m a matriarch, and a stable life helps me with my music. But right now, all I got in my brain is Vegas.”
“I wanna f****** nail it.”
The Easy On Me singer said of her upcoming Las Vegas residency: “I want to tell the story of the beginning of my career to now.
“I’m not gonna give too much away about it, but the show grows. It’s all about the music, and it’s really, really nostalgic.
“It’s gonna be really beautiful… I haven’t got any bullet points, obviously, when I talk. That’s why my shows are so erratic. One time I gave, like, an hour speech about an aquarium!”.
Adele made a return to the spotlight in 2021 with her fourth album, 30, launching it with a TV special, An Audience With Adele, recorded at the London Palladium.
It was filmed in front of an A-list audience, with guests including Dame Emma Thompson, Jodie Comer, Samuel L Jackson, Years And Years star Olly Alexander and Boy George.
It followed her interview with American chat show host Oprah Winfrey for a two-hour special, which was broadcast on American network CBS the week before.
The October issue of Elle UK is on sale from September 1.