There is a word for that !
... when it comes to love, there is a word for almost everything in French (there can never be enough words for love).
This is from the book - Benjamin Franklin: An American Life, describing Fanklin's relationship with his lifelong friend Catherine Ray -
"A romance? Yes, but a romance in Franklinian manner, somewhat risque' - somewhat avuncular, taking a bold step forward and an iconic step backward, implying that he is tempted as a man but respectful as a friend. Of all shades of feeling, this one, the one the French call this amitie amoureuse - a little beyond the platonic but short of grand passion - is perhaps the exquisite."
A little search revealed that it is also mentioned in the book 'The Private Life of Benjamin Franklin', where the author explains further - "It flatters but will not lead to heartbreak ...with a warmth, a sophistication, a sureness of touch, an intensity of personal interest that few men could match. But given a little time, a little reflection, he would pull back. He pulled back gently, not in a hasty retreat nor in a disavowal of feeling that would leave the girl bewildered and hurt."