I know that to make friends with somebody is idiomatic in English.
But in India I hear people saying
I want to make friendship with you
I think native speakers do not accept make friendship with somebody. I do not know whether it is grammatically correct or not.
But I think the following sentences are grammatically correct.
- I want to be your friend.
- I want to start friendship with you.
I want to know know other possible expressions with regard to make friends
I herewith attach a link which shows the usage.