This question does not have an absolute answer because there are always problems in defining the limits of cities.
The metropolitan area of New York City covers parts of New Jersey. (and possibly Connecticut if you cast your net wide enough), but the Jersey suburbs are not officially part of NYC. Similarly, Washington extends into Virginia and Maryland and greater St Louis extends across the Mississippi from Missouri into Illinois. I’m not sure how far the outer suburbs are considered part of the main city in either case. I note that the postal address of the Pentagon is in Washington, DC, although geographically I believe it’s in Arlington, VA.
Kansas City is
a traditionally accepted answer to this question, as it clearly spans both Kansas and Missouri.
The Kansas City metropolitan area is a bi-state metropolitan area. ... The larger Kansas City Metropolitan Area as seen on a map can be visualized roughly as four quadrants:
The southeast quadrant includes Kansas City, Missouri, and surrounding areas in Missouri. It includes the notorious Grandview Triangle.
The southwest quadrant includes all of Johnson County, Kansas, which includes the towns in the area known as Shawnee Mission, Kansas. Interstate 35 runs diagonally through Johnson County, Kansas, from the southwest to downtown Kansas City, Missouri.
https://en.wikipedia.org/wiki/Kansas_City_metropolitan_area
This website lists other towns that are considered "border towns/cities" -- i.e. towns that may be considered as valid for this question. Whether or not these count depends on your definition of "city" and how close / what parts of each city cross into another state. In some places a town takes on a new name when it crosses a border. So it it now two different towns?
Here are the candidates:
https://en.wikipedia.org/wiki/Border_town#United_States_within
[ This top question has been modified heavily by Terry, as it is a frequently asked question ]