FFDream.com have conducted an interview with Julien Merceron, the worldwide director of technology within Square Enix. The majority of this interview discusses their new "Luminous Engine", which was previewed at E3 2012, however, the interviewer asked about Kingdom Hearts III and if it will use the Luminous Engine, and Julien gave his answer.
It seems to suggest that some work has already been done with Kingdom Hearts on the Crystal Tools engine (the engine that Final Fantasy Versus XIII is built on), however, it could just mean that this is still in the planning stages, or it could simply be Julien's opinion. We'll let you draw your own conclusion from the interview extract below, which was translated from French to English by TheApprenticeofKingMickey from KH13.
Julien: It's already evident that we're using Luminous Engine on various franchises other than just Final Fantasy. However, Crystal Tools, as I was talking about earlier, is employed exclusively on Final Fantasy, only for now. Since the next Dragon Quest X uses Crystal Tools, as well as internal projects such as Kingdom Hearts which also utilize Crystal Tools, it's evident that Luminous Engine will cover a large spectrum of gaming franchises and I hope it will be taken advantage of by other teams, but we still have a lot of work to do.
Update: FFDream have done their own English translation of the interview, and this alternate translation can be read below.
Julien Merceron: We will use the Luminous Engine for a lot of games, not only Final Fantasy. By the way, currently, Final Fantasy uses the Crystal Tools, but also the news Dragon Quest X and other projects such as Kingdom Hearts. Luminous Engine will thus be used for a large range of games and, I hope, by a lot of teams, but we still have a lot of work to do.