When you say May I use your pen the word may is implying that you are asking for

In English grammar the modal verb may is often used to express permission. For example May I use your pen is a polite way of asking if the speaker has permission to use the object. Modal verbs like may can express possibility necessity obligation and probability depending on context. In this sentence it is not request direction or choice but clearly permission. Understanding modal verbs is important for both writing and speaking because they add precision to communication. May has a formal tone compared to can which is more casual. Therefore the correct meaning in this sentence is permission.