It's a little bit stilted. Perhaps most natural and polite would be: "Are you available for an English lesson today?" or "Would you be free to train me on English today?" I think "invite" is usually used if you are the one hosting. Since he is hosting the lesson, it makes more sense to politely ask about his schedule. You are making a "request" instead. But something like "I request you to train me on English today" still sounds very direct, so it is typically better to phrase your request in the form of a question.

I want to have an English lesson with my friend today and I am going to write to him this "Would you like to study English with me today." Is this correct? Thank you.