According to MacOtakara, (via AppleInsider), production of the iPhone 5S is already underway at Foxconn. As the design will essentially be the same as the iPhone 5, production lines won’t have any issue ramping up supply (Google Translation):

According to the informed sources who can trust it, Foxconn is talking with it seeming that I started a part of production of iPhone 5S at the plant.

It seems that most production-process alterations cannot be found so that it may be produced with the same line as iPhone 5.

Earlier today, it was reported the iPhone 5S would possibly launch in August, with an improved processor and camera upgrade, on par for a typical “S” upgrade from Apple.

iPhone 5S is rumoured to sport a fingerprint sensor underneath the Home button and a new ‘smart flash’ utilizing white and yellow LCDs for better pictures. The same reports speculate a lower cost iPhone could also launch alongside the iPhone 5S this summer.