- Could you tell me a ( near - nearby - close ) pharmacy?
- I live ( near - nearby - close ) to the bank.
- Don't leave. I'm (near - nearby - close ).
Do you think that they all are interchangeable here?
Any of those words would be suitable to convey the meaning of what you are trying to say, but there are subtle differences.
The words near and close are interchangeable, however the difference is that close implies a direct proximity, whereas near conveys more of a being in the same general area.
To show the difference:
A stranger was standing too close to me.
With no other information, this would make one think that the stranger is almost touching the person, and invading their personal space.
A stranger was standing too near to me.
This would be used more to describe someone who might be blocking the person's view.
The difference between them and nearby depends on who or what you are saying it in context to.
Using near/close would be if you are referencing the speaker/narrator, whereas nearby would be used if you are referencing the other object.
To use your examples, the correct words to use would be:
Could you tell me a nearby pharmacy?
This is used because you are speaking with the pharmacy as the subject.
I live near/close to the bank.
This is used on this occasion because the speaker is the subject, and not the bank.
These would not be used the other way round, unless you changed the subject of the sentence.
Am I near/close to a pharmacy?
I live here. The bank is nearby.