The Hunger Games: Mockingjay Part I will reportedly begin production on September 16th, according to a report from Production Weekly. The exact tweet from Production Weekly [@prodweek] read, "THE HUNGER GAMES: MOCKINGJAY PART 1 is scheduled to begin production September 16th with locations in Atlanta and Boston."
Production Weekly always been a reliable source of information, but nothing can technically be confirmed until it comes straight from the source. A misleading tweet from news anchor Rus Bowen a couple of weeks stated that Mockingjay will start filming in July, which turned out to be false.
Also, although the report states that only Mockingjay: Part 1 will begin filming, the studio and the producers have previously stated that they will likely shoot Mockingjay Part 1 and 2 together, much like Breaking Dawn was shot. The film will be split in post-production, but filming will take place together to keep the production cost from getting too high.
The second film in the Hunger Games franchise, 'Catching Fire' releases this November, and Mockingjay Part 1 is set for release on November 21st, 2014 and Mockingjay Part 2 is set for release on November 20th, 2015.