So Stefany and I have talked about this a couple of times. I come from the Chicago camp and she comes from the NYC camp. Here is what we found.
The distance between Gotham City and Metropolis has varied over the years, ranging everywhere from being hundreds of miles apart to being twin cities.
Why it is NYC
"Gotham" is a nickname for New York, first used by Washington Irving in the early 19th century.
For most of the publication history of Batman in comics, Gotham has been assumed to be a New York City analogue; Frank Miller has said that "Metropolis is New York in the daytime; Gotham City is New York at night."
DC Comics president and publisher Paul Levitz says that Gotham is "New York from 14th Street down, the older buildings, more brick-and-mortar as opposed to steel-and-glass."
New York Times journalist William Safire describes Gotham City as "New York below 14th Street, from SoHo to Greenwich Village, the Bowery, Little Italy, Chinatown, and the sinister areas around the base of the Manhattan and Brooklyn Bridges.
Tim Burton's Gotham was based primarily on New York,
"Batman Forever," (1995) was filmed partly in New York City,
In the Batgirl series, as well as in the Vertigo Comics' Sandman series, Gotham is implied to be an entire state, analogous to New York, with Gotham City as its capital. In both cases, the book refers to "Upstate Gotham"
Why it is Chicago
Longtime Batman artist Neal Adams considers Gotham to be based on Chicago (whose nickname is "New Gotham"), pointing to its history of corruption and organized crime, and adding, "One of the things about Chicago is Chicago has alleys (which are virtually nonexistent in New York). Back alleys, that's where Batman fights all the bad guys."
Films directed by Christopher Nolan have shown a Gotham more closely based on Chicago.
Man-Bat #3 refers to Gotham City being in the Central Time zone.
"The Dark Knight", (2008), the film depending most on exteriors, was filmed entirely on location in Chicago, including recognizable shots using the Sears Tower, the Trump Tower, LaSalle St. and Navy Pier (for the Joker's seemingly fatal joke involving the two boats).
In The Dark Knight, the License plates of the cars registered in Gotham strongly resemble that of Illinois but they say Gotham for the state.
However there is a third place that it could be.
Why it is New Jersey
A Gotham City driver's license shown in Batman: Shadow of the Bat annual #1, contains the line "Gotham City, NJ".
A location on the Jersey Shore is described as "twenty miles north of Gotham." Robin and Batgirl drive from a "secret New Jersey airfield" to Gotham City and then drive on the "Hudson County Highway". Hudson County is the name of an actual New Jersey county.
New Adventures of Superboy #22 (October 1981), placed Gotham City and Metropolis on opposite sides of a Delaware Bay, with Gotham City in New Jersey and Metropolis in Delaware.