The J. Edgar actor was on a movie shoot in Big Sur, California with a mystery girlfriend when she informed him of her premonition that they were about to be attacked in their trailer by a cougar.
"I'm like, 'Honey, this is Big Sur, there are no mountain lions'," he told Jay Leno on Thursday's edition of the Tonight Show. "[The next day], I go home at four o'clock in the morning and realise I don't have a flashlight.
"I get to the point where I realise I can see nothing... I hear, totally serious, two big breaths, and six feet above me, I see the very slight glint of the eyes of a huge [mountain lion].
"I started screaming and yelling and literally being as crazy as I could be, it padded off and I worked my way into the trailer. I decided at that moment that this woman was the woman for me."
Lucas has been romantically linked to stars such as Anne Hathaway, Rachel McAdams and Ashley Olsen in recent years.
J. Edgar is released in the UK on January 20.
> Josh Lucas: 'I'm a secret father figure'
Watch Josh Lucas in the trailer for J. Edgar below: