These next iterations of the iPhone are expected to launch in the fall, with speculation targeting the September timeframe. 4.7-inch and 5.5-inch displays have been mentioned before, and the source "confirmed" those sizes, saying the smaller version will likely be produced first, while the larger version could be delayed. Japan Display, Sharp and LG Display have been selected to make the screens, said the sources.
It's been long speculated that Apple's next iPhone -- the iPhone 6, as this is a "non-S" year -- would feature a larger screen to catch up to Android and Windows Phone devices, which go up in size past 6-inches in "phablet" form. It's been also rumored that there would be two models this year, with one featuring a 4.7-inch screen and the other a 5.5-inch screen.
The reason that the larger screens would trail the 4.7-inch version is apparently Apple's use of in-cell touch panel technology. That tech allows for thinner touch panel construction, and was introduced with the iPhone 5. However, the sources said, due to difficulties with in-cell production technology for the 5.5-inch size, a decision has been made to begin mass production on the 4.7-inch version alone.
Production of 5.5-inch screens is expected to begin several months later, and it is possible that Apple might be forced to shift to a film sensor instead of in-cell technology, a source said. "Several" could also mean that the 5.5-inch size might come in 2015, rather that 2014, but this is all iPhone-speculation.