The report comes from two different sources. One is a report from BrightWire News, which cited supplier sources. Commercial Times, via Bloomberg, also reported the rumor.
BrightWire News said:
[Apple] halted processor production for [the] iPhone 5S in May.The iPhone 5S was expected to hit the market in September or October with the same 4-inch screen as in the iPhone 5. Both reports say that it has now been delayed until the end of the year. The so-called budget or lower-cost iPhone will still be released in October, the reports said, and its design has not been changed.
This is, of course, just another rumor in the treasure chest the continues to reach the eyes and ears of media almost daily. There are some reasons to look askance at this one, though.
Apple's hammered stock doesn't need a delay of the iPhone 5S. Apple is well aware that it needs a new flagship device, and soon.
Apple said, previously, that iOS 7 was coming in the fall. While it's possible iOS 7 could launch prior to the iPhone 5S, it's doubtful. It's also doubtful a budget model would release prior to Apple's new flagship model, but again, it is possible.
As MacRumors pointed out, the claim is unlikely to be true, as Apple almost certainly finalized the iPhone 5S design months ago.
S models are generally modified only slightly from their older siblings, with most of the modification being done internally. A change to the screen size would deviate from that formula.
Of course, the size specified as the new iPhone 5S screen size, of 4.3-inches, is a pretty popular one nowadays. It's big enough to offer good visibility to those wanting a larger screen and larger font, but not large enough to be in the phablet range, like the Galaxy Note II.