It's a little bit stilted. Perhaps most natural and polite would be:
"Are you available for an English lesson today?"
"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.