LG announced their first-gen Android Wear smartwatch (the G Watch, which uses an LCD display) in March 2014, and the company is already reportedly getting ready to release the 2nd gen smartwatch next month during the IFA 2014 trade-show.
According to Korea Times, the new G Watch 2 will adopt an OLED display, produced by LG Display. If this is true, it's highly likely that this is a flexible OLED. After all, LG Display does not produce non-flexible small OLED panels.
The Korea Times also reports that LG Chem will provide the battery for this watch. LG Chem was the provider of the G Flex curved batteries, by the way, which is another sign that if this report is true it's probably a flexible OLED.
The report says that the G Watch will be LG's answer to Apple's upcoming iWatch. It is estimated that Apple will also adopt a flexible OLED for the iWatch - and LG is rumored to be the exclusive supplier (although Apple may be approaching Samsung Display as a second flexible OLED supplier).
LG Electronics already has a wearable smartband that adopts an OLED, the LifeBand Touch with its small non-flexible white PMOLED panel. Samsung's Gear Fit, meanwhile, uses a flexible 1.84" (432x128) Super AMOLED panel.