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.
AND
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.
2This is not in common usage--I can't think of an example when I would use might or may this way. Use "He told her that she could borrow his car" instead. – farnsy – 2017-12-22T01:16:29.357
Your question isn't clear. Do you want to ask the meaning of may in this context or do you want to know what's the indirect form of your sentence? If it's the latter, B is the proper answer but I think @farnsy is right. It's not a common usage. – Aragaki Aya – 2017-12-22T01:47:43.533
I want to ask which option is most appropriate , A or B. – asr09 – 2017-12-22T01:49:25.123
"He permitted her to borrow his car" is fine. – farnsy – 2017-12-22T02:17:46.717
I realized reading over this again, that, in fact, "may has two uses in English. I have tried to sort them out in my answer and explain them. It took rereading the question several times for me to figure out why I was uncomfortable with might. I had sort of forgotten all this from my English teaching days... – Lambie – 2018-05-10T18:43:57.757