Game director and producer Naoki Yoshida (also known as Yoshi-P) was recently asked by Easy Allies about the possibility of a Nintendo release and mentioned how he wants to see the game available on as many devices as possible in the long run. This apparently includes Nintendo's hardware, and at the rate it's going, the new target could potentially be the Switch's "successor".
According to Yoshida, it's a longtime "dream, or goal" for him to allow players to be able to play this online Final Fantasy game on any device, and a release on a Nintendo platform in the future is something he would absolutely "love" to see happen. Here's exactly what he had to say (via Nintendo Everything):
"My dream, or goal, for more than ten years at this points is for players to be able to play Final Fantasy XIV regardless of device, and be able to access the servers to go in and access the same world. … we've been talking for years to bring the game on the Xbox platform, and so of course we would love to have our title available on the Nintendo platform, and we do want to continue our efforts. But it did take a lot of time – with Xbox, we literally have talked for years, and it is now finally coming to fruition. We will try our best and continue to work hard at it and continue our efforts. Eventually, it would be cool – something that we would love to sort of explore and make a reality one day."
Again, it seems like we probably won't be seeing Final Fantasy XIV Online during this current Nintendo generation. Even if it did happen, based on certain other releases, it could be locked to select regions or distribution methods such as cloud streaming. E.g. Dragon Quest X Online (exclusive to Japan) and Phantasy Star Online 2 (a cloud release on Switch).
Nintendo has also mentioned now how its Switch "successor" will be revealed this fiscal year, so this comment is likely already taking into account future hardware releases from Nintendo.
Final Fantasy XIV Online was most recently released on Xbox platforms, with players able to enjoy a free trial of the game which offers quite a large amount of content.
If you're not familiar with Final Fantasy XIV Online, as the name suggests, it's an online version of the numbered series allowing you to explore role play as a unique character, take on a variety of classes and jobs, live out your story, and save the day alongside a player base of more than 30 million adventurers.