1
general::
locally
in or near the area where you are, or the area you are talking about:
• The restaurant only uses vegetables which are grown locally. • Police say they believe the man lives locally.
nearby/close by
not far away:
• Do you live nearby? • The villa has a private garden and there’s a swimming pool close by.
around here
spoken in or near the place where you are:
• Is there a bank around here? • Do you live around here?
in the neighbourhood
British English , in the neighborhood American English in the area of the town where you are, or that you are talking about:
• Are there good schools in the neighborhood?
in/around these parts
in the particular area of the country where you are, or that you are talking about:
• There aren’t many foxes in these parts. • He’s known around these parts as the ‘bag man’ because of the bag he always carries with him.
Longman-Thesaurus