Yes, those feelings are a "normal" part of therapy although personally, but I just can't see that sorta close but not close being healthy... humans are social beings. Despite this, I am still in therapy with a T that will never be anything more than a T.
I have had the "can we be friends" conversation with 4 T's... Got 2 yeses that were really no and did a lot of harm, got one flat out no (and it wasn't as hard to hear as I thought it would be)... and one very precious yes.
So... Don't like it, don't agree with it... but as long as your T is strong enough to be honest with you it is good to talk about. My current T (the flat out no) had a really wonderful way of explaining it... So... I hope that if the answer is no that your T will have just as supportive reason as mine did.