The race is on to get the first TV interview with Jaycee Dugard, the young California woman who was kidnapped at age 11 and held captive for 18 years by a sexual predator.

Dugard, who yesterday announced that she had written a memoir about her ordeal that would be published in mid-July, is suddenly the prize in a high-stakes poker game that features the biggest names in TV news — including Diane Sawyer, Tom Brokaw, Anderson Cooper and maybe even Katie Couric, according to sources.

A publicist for the now-31-year-old Jaycee — who had two children by her kidnapper, Phillip Garrido — has scheduled meetings with the big players in LA beginning next week.

But the final choice of interviewer will be “up to Jaycee,” says her spokeswoman, Nancy Seltzer.

Tentatively, the interview will air sometime in early July, before the July 12 publishing release. “But it will be on her timing,” says the spokeswoman.

Money could play a major role in her decision.

A home video of Jaycee with her sister and mother, recorded shortly after she returned home, was sold to ABC in March 2010 for more than $200,000.

The video and an interview with Jaycee’s mother, Terry Probyn, appeared on “20/20.”

Jaycee’s family has been taking videos of her in the 13 months since the ABC appearance, says Seltzer, and “possibly” the right to the footage will be part of the deal for the first interview.

CNN, CBS, ABC and NBC all claim that, as a matter of policy, they do not pay for interviews.

But, in practice, there are ways around that restriction.

In the past, network news departments have found ways to get money to newsworthy subjects by paying inflated fees for the rights to use photos and videos or to rent the house where the interview is taped.

Still, the first primetime interview with Jaycee is likely to be a ratings bonanza and sharing some of the millions in ad revenue with Dugard is only fair, her reps contend.

The courtship of Dugard went into high gear in the last few weeks since Garrido and his wife Nancy, a nurse’s aide, pleaded guilty to the kidnapping in plea deals that meant Jaycee would not have to testify.

Couric has been writing to Jaycee since last year, seeking an interview “when the time was right,” according to sources.

Now that she is leaving CBS News and apparently headed for ABC, nothing would prevent her from taking the interview with her — if she signs there quickly.

Spokespeople for all three networks declined to comment for this story.

It has not been decided whether Dugard will do more than one interview, says Seltzer.

The only thing that is for sure, the publicist says: the TV appearance will not include her children, Angel, 16, and Starlit, 12.

“The kids are minors and are off limits,” Seltzer says.