He is gross and creepy.
You worry that you haven't told him 'no' forcefully enough - I would bet that he knows he is being inappropriate and making you uncomfortable but is hiding behind the idea that since you haven't directly told him "I don't want to talk to you" he can pretend he "misread signals" or "is just being friendly".
If you are considering reporting it, it would be very useful for your case to have a very clear, unambiguous example of you telling him to stop. If he stops = yahtzee! Problem solved. If he ignores that or gets wounded/shirty/angry then you will be in a strong position to report him.
Send him a message that says "Look, I don't feel comfortable with you sending me messages or having contact with me outside of lessons/unrelated to the course. I think it's a bit inappropriate, since you are my tutor and everything. I'm sure you agree - see you in class next week/whenever"
Then don't respond to anything he sends, but do take screenshots and document anything he says to you in class. Ask if you can record the sessions if you think that would be OK - pretend you want to write up notes later or something.
My guess is that he will respond with a mixture of bafflement and denial ("Wait, you think that there's something....sexual....somehow about me messaging a student to say what she would look like lying on my bed?? That's so sad you would think that. And, I'm sorry to say, a little immature of you. I'm disappointed" etc) and go on about how sad he will be to not have your chats, how he was just "being friendly" and you have taken it all the wrong way, blah blah, blame you, etc.
Ignore him. Send that last message, then don't respond to ANY further ones, even if he sends one that seems like a crisis about your course or something to do with your work, unless it comes through proper channels.
In class, be brisk, cheerful and professional - and if he starts to make things weird for you, tell your classmates (you might want to tell them anyway) so they can witness any retaliatory behavoiur and corroborate it.
It sucks you have to do all this because a creepy guy is using his power to harrass you, OP. None of this is your fault - this is 100% not OK.