Apple is currently relying on Samsung Display as its exclusive supplier of smartphone OLED displays (SDC produces the 5.8" flexible Super AMOLED display on its flagship iPhone X), and Apple is obviously not content with this situation.
Finding a second supplier is not easy as SDC is the only company that can produce quality flexible OLEDs at the quantities that Apple requires. Reports suggested that Apple is in talks with LGD (also with other companies, such BOE Display) and in its recent filing, LG Display hinted that it is indeed talking to Apple.
LG Display noted that "Regarding the OLED supply deal for Apple's iPhone X, nothing has been set in detail, when anything is confirmed in detail, we will announce it...".
This is of course not a confirmation, and anyway it is likely that Apple is in discussions indeed with all OLED producers, but this statement may mean that LGD is closer to an OLED supply deal to Apple than other companies (which makes sense as it is also closer to volume mass production).
LGD has recently started mass producing flexible OLEDs at its E5 6-Gen production line, but volumes are still low (LGD probably only produced around 1 million smartphone OLED panels in Q3 2017). LGD is currently shipping a single smartphone AMOLED panel, a 6" 1440x2880 (537 PPI) one used in LG's own V30 and Google's Pixel 2 XL. Reviewers and consumers alike were disappointed in LG's AMOLED performance, which means that LGD will need to step up its game quickly if it wants to win orders from Apple for 2018 or 2019.