"I'd sew the hems of his pants if he asked me to," Johansson, 22, said in the Vogue interview, USA Today reported Tuesday.
For his part, Allen, 71, called his "Match Point" star "criminally sexy" in an e-mail to Vogue.
"She is unlike anyone who has come before her, and while she is a much stronger actress in every way, there is a tiny bit of Marilyn Monroe in her zaftig humidity," he wrote.
The Johansson-Allen attraction is purely professional, she said, telling told Vogue she works to keep her romantic life private.
But Johansson said reports last month that she and singer Justin Timberlake were an item were "not true" and when asked about "Black Dahlia" co-star Josh Hartnett, she offered, "I'd rather not comment on my personal life in that way."
Johansson's next film, "The Nanny Diaries," opens April 20.
Celebrity Families of 2014 [PHOTOS]