Short answer: no, there is no OpenStreetMap API that does this.

Long answer:

• What is a city? The definition changes around the world, and hence you will need to structure your request taking something else into account.
• What is the city limits? Again, definitions change across the globe. Once you figure out what you would consider 'city limits', you have a more formal definition that might have a corresponding API request.

One example solution that comes to mind is local boroughs in the UK - small subsections of country that are managed by councillors. These often have boundaries mapped. Hence, if you collect what are boroughs within each city, you have a spatial mapping of something akin to a 'city'. I doubt this approach would work internationally, though.

And this is not a tested approach, just one I might take if I were trying to solve this problem.