May asks permission and sounds more formal; could handles polite requests and possibility (The delivery could arrive Monday), though may also works for that. In a permission ask, both are fine: May I leave early? is a touch more official than Could I leave early?.
Paired sentences
May I leave early on Friday? and Could I leave early on Friday? Both ask permission politely. May sounds a little more formal, which suits a note to a manager you barely know. Could reads as friendlier, which suits a manager you message often.
The delivery may arrive on Monday. and The delivery could arrive on Monday. Both say it is possible without committing to it. If your sentence is about what might happen, either word is a fair pick; pick by ear, or by how formal the message needs to feel.
Could you send me the draft tonight? and May you send me the draft tonight? The first is the natural request in a message. The second sounds odd, because may asks about permission, not action, and the sender is not in a position to grant it.
Overlap worth knowing: in a permission question to someone with authority, either word can work. May I borrow your projector? is more formal. Could I borrow your projector? is everyday-polite. Choose by how distant or official the exchange feels.
The reply, not the request
One trap: the word that works in the request does not always work in the reply. If a colleague writes Could I use the meeting room?, answer Yes, you can or Yes, you may. Replying Yes, you could sounds hesitant, as if the room only might be available.
So when you are the one granting or refusing, reach for can or may. No, you can't borrow it today is a normal refusal. No, you couldn't does not read that way. The asymmetry is small but real: could asks, can and may answer.
Which word you choose depends on how formal the situation is. For formal permission, grant with may: Yes, you may submit the form next week. For everyday messages, can does the work of granting or refusing: Yes, you can come along and No, you can't borrow it today both sound natural. Save could for the request itself.
- A fresh example of formal granting: Yes, you may submit the form next week. In a casual reply, the same permission sounds like Sure, you can send it whenever. Both grant permission; may keeps the reply formal, while can suits everyday messages between people who know each other well.