Goodspeed was found dead of a suspected suicide outside "American Idol" judge Paula Abdul's Los Angeles house last month.
Abdul has since described Goodspeed as a stalker. Abdul has also claimed she was nervous and told "Idol" producers three years ago when Goodspeed arrived to audition that she had had trouble with Goodspeed in the past.
But Lythgoe told "Access Hollywood" in an interview to air Friday that he and the show's other producers weren't worried about Goodspeed auditioning.
"The producers had seen her, we'd seen her, she was an absolute fan of Paula's, so she's going to go in there to say, 'I am your biggest fan,'" Lythgoe said. "Now that gives Simon Cowell great material, which if you saw it was all things like: 'Do you sing as well as Paula?' and 'Do you copy Paula?' If we had known that she was in any way, shape or form, a danger to herself or Paula, there is no way we would have allowed that."
Lythgoe also said he doesn't think it's fair for anyone to connect Goodspeed's apparent suicide with her failed bid to compete on the show three years ago.
"To sort of tie that to an audition on 'American Idol' I find ... upsetting, and for it to continually to be raked over, I find upsetting for her family," he told "Access Hollywood."