Companies vying for the four billion dollar (3.2 billion euro) contract include Raytheon and Lockheed Martin of the United States, Rosoboronexport of Russia, CPMIEC of China and the French-Italian consortium, Eurosam, they said.
A Western diplomatic source said a shortlist would be announced on July 11 or 12.
The news comes a week after Syria shot down a Turkish warplane, prompting Turkey to start beefing up its forces along their lengthy border with anti-aircraft guns, missile batteries and tanks.
The Western diplomat said France had a better chance of winning the order following Francois Hollande's May election in succession to Nicolas Sarkozy, who upset Turkey with plans to make denial of the Armenian genocide a criminal offense.
China and Russia are unlikely to secure the contract for Turkey, a NATO member, since they are not part of the alliance, Hurriyet Daily News reported.
Turkey also hosts a NATO early warning radar as part of a new defense system aimed at protecting Europe from missiles coming from the Middle East, notably Iran.