Asked whether he is still engaged to make-up artist Mezhgan Hussainy, 37, he replied: “I’m not sure,” fuelling reports their romance may be over.
Stern then said: “You seem embarrassed by the women you date, and I’ll tell you why, they are lovely women, but yet you won’t talk about about them.
“I’m a married man and if someone asks me a question about her I am proud to admit I have a relationship.”
Cowell replied: “If we were having this conversation off the radio I could have a normal conversation.”
But Stern insisted: “Why is the question ‘are you engaged?’ so painful to you?”
An irritated Cowell responded that it “wasn’t a pretence of an engagement” and that he wouldn’t be discussing his relationship any further.
During the interview, the millionaire also revealed “bizarre kind of blind date” with Charlie Sheen’s ex Denise Richards, 40, when she was eight months pregnant, mistakenly calling her Denise Lewis before being corrected by Stern.
“I did have a bizarre blind date with Denise Lewis, Charlie Sheen’s ex,” he said.
“She’s cute. We met in a hotel and she turned up eight months pregnant with a dog in her handbag. She’s a bit of a whacko, but I like that.”
The actress reacted to the news on Twitter, saying she didn’t see their meeting as a date.
“Unbelievable…what’s wrong with people,” she wrote. “I was at a group dinner 8mo pregnant not sure how thats considered a date, news to me! If I was being set up I didn’t know it!”
“S**t I wish I knew that it was a date!!! He’s a hot piece of a*s!”
In the same interview, Cowell also admitted having a threesome “years ago”: “I was just with these two girls and we ended up in bed. It was cool, I liked it.”