Could I borrow your slides? asks politely, casually. May I borrow your slides? means the same but sounds more formal, right for a boss or stranger. With a colleague or shop assistant, either works, so pick what feels natural.
Side by side in a message
:- Could I leave early today? The friendlier everyday option, what most people send without thinking. May I leave early today? The same request, but more formal and respectful, better for a strict boss or official letter.
The client may reply by Friday. This says a reply is possible. The client could reply by Friday. Also possible. For possibility the two sit very close, so either version keeps your meaning intact.
Could can add a condition, like a route you are imagining. The client could reply by Friday if we nudge them. Here could suggests what might happen under one condition, a nuance may does not carry as naturally.
One more pair, kept short: Could we move the meeting to Thursday? is polite but casual, fine for a teammate in chat. May we move the meeting to Thursday? lands more formally, which can feel stiff in a quick message.
Where the choice actually flips
One caveat that can change your choice: when you answer a request rather than make one. Could fits naturally on the asking side, but as a granting reply it gets shaky.
Someone asks May I send the file now? A formal reply is Yes, you may. Plain Sure also works.
Replying Yes, you could reads odd on its own. This is a tendency rather than a strict rule, and an informal Yes, you could does show up in some usage tables.