city centre vs city center

city centre and city center both are nouns.

Word	Noun	Adjective	Verb	Adverb
city centre	Yes	No	No	No
city center	Yes	No	No	No

As nouns, city centre and city center are synonyms defined as:

Other synonyms of city centre include central city.

city centre (noun)	city center (noun)
the central part of a city	the central part of a city