There are two uses of **may in English**
1) - To ask permission as in "May I borrow you car?"
Answer: Yes, you may; No, you may not.
2) - May is a modal verb used to state something that is not 100% certain: [He said:] "You may borrow his car tonight."
The reported speech for 1) would then be:
He said she could borrow his car. Or:
He said she could not borrow his car. [AKA He gave her permission to borrow his car.] This could even be: He allowed her to borrow his car. BUT: it cannot be might.
The reported speech for 2) in the present tense would then be:
He said she may borrow his car tonight.
Past Tense: He said she might borrow his car tonight.
In 2), may is used to express uncertainty about whether she will or will not borrow his car. In 1) one person is being asked permission by another to borrow something.