Vatican City does not have a city in it, but is in another city, the city of Rome. And, by the way, Vatican City is also a country, so the city of Rome has a country within it.
The City of Pittsburgh (PA) surrounds the city of Mount Oliver.
In Australian cities, particularly Brisbane, Sydney and Melbourne, there are what we call City Councils. Depending on what you define a a city, then they could be considered as cities.
Manchester, UK has the city of Salford practically inside it. Also London, UK has the area inside it known as the City of London, although it is just a region of the city. Croydon is also a city within the Greater London area.