Gotham's choice to set itself apart from the main comic universe seemed to be in place from the start, with important characters like Fish Mooney not even existing in other Batman stories. However, this divide from the comic universe led to many questions and theories regarding if Gotham could be setting up a universe we were already familiar with, rather than a completely new one. In my personal experience, I have seen many theories predicting that the show exists as a prequel to several universes, including those of Christopher Nolan's The Dark Knight Trilogy, The DC Extended Universe, and Batman Beyond. However, one universe with plenty of material seems to line up with Gotham, almost a little too much to be untrue, that of Frank Miller's The Dark Knight Returns.
We'll begin with the heart of any Batman story, Batman himself. There are a lot of relationships developing at a young age for Bruce on Gotham that show up again in The Dark Knight Returns, such as those between Bruce and Selina, later, Catwoman, as well as Bruce and Jim Gordon. The closeness of these in the original graphic novel seem to have something behind them, more than just years of crimefighting together. We see this connect as Bruce begins and grows these relationships from a very young age, should the timelines be connected.
SPOILERS FOR TDKR AND GOTHAM SEASON 2 & 3: Also, we have a fan favorite of both Gotham and TDKR, The Joker. Now, it remains to be seen if Jerome on Gotham will become the true Joker, but with the developments of the back half of Season 3, it seems to becoming to fruition. One key point that is limited to a few tellings of The Joker's story is the mutilation of his face. We see this in the New 52, but Batman and the Joker have an established relationship prior to that, which rules out any relation to these stories. Now, as far as we know, the Joker's face is generally undamaged in TDKR. However, he seems to have a likeness not unlike a recipient of many expensive plastic surgeries. As we see multiple times throughout TDKR, Dr.Wolper, as well as many other civilians and 'higher-ups' are invested in Joker's rehabilitation and assimilation into society. A significant plot point throughout the story of TDKR is the idea that the rehabilitation of one of the world's most dangerous psychopaths is no small feat to accomplish and would bring much renown to anyone responsible. How do these points all come together? Well, we see that Jerome's face has been severely damaged, borderline removed as of his return in Season 3 of Gotham. Should he continue his career as The Clown Prince of Crime, and end up in a psychiatric facility, many benefactors would no doubt be willing to pour money into the cause. This would eventually lead to many plastic surgeries to repair Joker's facial wounds, leaving him with the stiff look we see in TDKR. END SPOILERS
One of the biggest complaints and confusions about Gotham as a series is the age gap between Bruce Wayne and iconic villains we are led to believe he will face in his future as Batman, including The Penguin and The Riddler, most prominently. A 'convenient' connection to The Dark Knight Returns is that the Penguin and Riddler are completely absent and unmentioned in all continuations and volumes of the series. This not only allows, but cultivates the belief that Batman's adventures against these adversaries played out in the early days of his career, due to their many years over him. Neither of these have ever presented physical threats to The Caped Crusader, so their age in the beginning of his career would not affect their ability to terrorize Gotham. Plus, it only adds inspiration to Bruce's reasoning for becoming the Batman if super-villains like these are already roaming the streets.
Of course, the Gotham-verse shares similarities and hints towards almost any Batman tale if one wishes to perceive it that way, yet some remain undeniable. Strong theories are built around the Tim Burton universe, the Batman Beyond universe, and the list goes on. However, based on my theory and the aforementioned proof, I believe Gotham leads directly into the Dark Knight Returns universe.