The film will star Bradley Cooper, who is developing the project as a producer, the newspaper said.
Cooper, 38, achieved fame with his roles in "The Hangover," "The A-Team," "Limitless" and "Silver Linings Playbook." His "Silver Linings" work earned him his first Academy Award nomination for Best Actor.
Production on the film, written by Jason Hall, who wrote the comedy drama "Spread," is expected to start early next year, the Reporter said.
The film will be a Warner Bros. Entertainment and DreamWorks Studios co-production.
Kyle's best-selling memoir, subtitled "The Autobiography of the Most Lethal Sniper in U.S. Military History," discloses how the Texas native, whose father taught him to shoot when he was 8, went from rodeo cowboy to expert marksman and feared assassin, with the highest number of sniper kills for an American.
Kyle, 38, and companion Chad Littlefield were shot and killed in February at a Texas shooting range by fellow veteran Eddie Ray Routh. Kyle and Littlefield were reported to have taken Routh to the range to help him with his post-traumatic stress disorder.