To feel attached to someone

I don't consider the phrase that strong. I am attached to my books, to my house, to my teacher.

So what expressions can I use to refer to a person?

If you love the person, use 'love'. If it is less than that, you can 'be very fond of' (him)' or 'like (him) very much'. No doubt other members will have other expressions to offer.

How about the expression "to grow attach to someone/each other"? Don't you use such expression to say that they are loving each other. (The process of falling love, but not exactly love. For example the love between two friends)

I believe that I have already expressed my feeling that this did not mean that. It does not have sufficient passion, to my mind. Others may have a different opinion.

How about "They were getting fond of each other"?

This is the idiom:

They were growing fond of each other.

And how about: "His heart ha been attached to her."

No, that is perhaps from your own language; it is not an English expression.

How about, "His heart had been tied to her."