Should I ask him or let him ask me?

you guys sound like you're already good friends. Maybe he sees it as you guys are already together; in a way. You could always ask him to have dinner at your house, or come to your house to study or something casual like that. He sounds like he really likes you, and thinks that you already know it. (It sounds really good for you two if both of you are currently single) You could even ask him (casually or even jokingly) what kind of girl he is interested in; and maybe you already fit the bill. He sounds really shy to me and that his friends are embaring him, so try to take that into consideration and go easy on him. He sounds like a good guy so good luck!