ff12 and ff14

[Enkidoh]

Except SE themselves have specifically addressed by stating outright that FFXIV has no direct connection to FFXII, FFXI or any other FF game (it was a post by Bayohne that stated this in a past 'Letter From the Producer Q@A' , originally found here which sadly seems to have disappeared, but I luckily saved the post in question and have included it here):

Q55: There are a lot of names from Ivalice such as Dalmasca, Belias, and Zodiak that have been mentioned, is there some connection between Hydaelyn and Ivalice?

A55: To be absolutely clear, there is no connection between FFXIV’s Hydaelyn and FFXII’s Ivalice. Likewise there is no connection with FFXI’s Vana’diel. Each are their own individual world. Based on this, there are connections between the things that appear in the world, and we’ll be revealing more in the future.

And yes, there is the Void in FFXIV that also appears throughout the FF series as a kind of 'world between worlds', but again, it doesn't mean FFXIV is directly connected to any other FF game. Gilgamesh's appearance in ARR has never really been explained other than a vague "I've found myself in your world" sort of thing, so until SE states outright that he arrived via the Void or by some other means, it is mere speculation.