"We have a lot of fun. He's, like, my best friend," Grande told Minaj. "It just gets better every day."
"We're like the boy and girl version of each other, except he's 17 feet tall and I'm 4 inches tall," she jokingly added.
Grande said her first kiss with the Saturday Night Live star was a special experience.
"It was so sick. It was so dope," the star said. "It was so cute; he asked my permission to kiss me."
"It wasn't, like, too naughty of a kiss," she added. "It was really sweet, but it was also really sexy."
Grande and Davidson confirmed their engagement in June after a few weeks of dating. Davidson told Variety in an interview published Tuesday that he proposed during a quiet moment in bed.
"I didn't want to do something corny," the actor said. "We were in bed hanging, after watching a movie. I was like, 'Will you marry me?' It was really dope."
"I never thought I'd meet anyone like her," he said. "I can't even put into words how great of a person she is. I could cry."
Grande is also celebrating the release of her fourth studio album, Sweetener, which debuted Aug. 17. She will film the music and interview special Ariana at the BBC in front of a studio audience Sept. 7.