The iPhone 5S is expected to be released early fall. Japanese Nikkan (via AppleInsider) reports that Sharp is set to start production of iPhone 5S panels in June. Sharp will not be the only supplier of panels for the iPhone 5S as it was for the iPhone 5. It is actually predictable that suppliers for the iPhone 5S will start making components in June and July.
The report is not making any references to the potential other iPhone Apple is rumored to release this year. It is also not clear if the iPhone 5S display is any better or different than the panel found in the iPhone 5S.