Before starring as the lead detective in the CBS missing-persons drama “Cold Case,” Kathryn Morris was courted by Steven Spielberg and Rod Lurie. She played every man’s fantasy in HBO’s “The Mind of the Married Man” and in John Woo’s “Paycheck.” Now, seven years into playing “Cold Case’s” Lilly Rush, Morris says she’s more content than ever.

Q. Spielberg sought you after seeing you in Lurie’s political thriller “The Contender.” He later cast you in “Minority Report” and “A.I.” How did being pursued by Spielberg feel?

A. You think it’s a joke when your agent calls you, and then all of a sudden you’re in front of him. It’s like getting sprinkled with fairy dust. After 10 years in the business, there you are sitting in front of him, and he’s got your picture and résumé on his desk. I remember he said that he thought I was a very versatile actress, and that’s why we were having this meeting. … It felt like he was bringing me into his club.

Q. Lurie, who cast you in “The Contender,” called you his muse. Adoration!

A. He’s another decent person. He actually saw me on an audition; it’s as boring as that. But we talk a lot. He has his own like little acting company, if you will, and I think he just feels like I’m one of his go-to actors. When he writes a new script, often he will send it to me. I feel honored.

Q. You’ve been playing Lilly on “Cold Case” for years now. How do you keep a procedural fresh?

A. Lilly has evolved from being this little beat cop to this hungry cold case detective to a person who has expanded her capacity. Not just with the suspects or the victims. She has really evolved into a person that is as real to me as a relative of mine. It’s a pretty cool thing to have for this many years.

Q. Have you felt restless or like trying a new role?

A. I’m in talks for a couple of great independent films concurrent with “Cold Case,” but the show is having its best season ever, if you ask me. To me, when actors get restless on series, I find them to be a little on the lazy side. I’m still looking to be a better actor, and I’m grateful every day.

Q. Before “Cold Case,” you had a juicy role in HBO’s “Mind of the Married Man.” Could you see doing comedy again?

A. Ha, that show really (ticked) some women off, didn’t it? It became a little controversial, and I loved being part of the controversy. Playing the press secretary to the mayor, being this fantasy of a smart bossy woman who can roll in lingerie and have sex while talking politics in bed ... I loved it. It was just so much fun, but again, I’m really happy on “Cold Case” now. It’s like doing a mini movie every week.

Q. In “Cold Case,” you jump around in time, solving missing persons cases. Do you have a favorite time period the show has gotten to explore?

A. I actually prefer cases that are a little bit more recent. But I do like the ’80s because of the music. In our two-part season finale, it was exclusively Pearl Jam. That was just the coolest.

Q. The time slot changes have not been helpful to the show’s ratings.

A. Well, we have fans that have followed us wherever we go. And our audience actually watches the show when it airs, they don’t DVR it. That’s a big difference. We are trying to do more cases aimed at 18- to 49-year-olds, though. Like we just did a skateboard episode, and a fight club episode.