phrase request

"Will you be my friend?" is the kind of thing children might say to each other. Adults rarely feel comfortable so openly asking for friendship. Alternatively, most people will suggest activities you can do together, for example:

Hey, let's go out and get something to eat sometime? I know this great restaurant.

or

Do you like noodles? I know this great restaurant.

Do you want to come over and hang out? We can watch TV or whatever. The game is on this weekend.

Have you seen that movie [X]? Why don't we go this weekend?

Why don't we grab a beer after work?

And so on. Of course, depending on your tone and other factors, a casual offer of friendship could be misinterpreted as flirting, but I'm sure that's the same in any culture.

Of course, some people are better at this than others, and more natural at making friends. If you are a friendly person, it often doesn't really matter what you say as long as your intention comes across.

Edit: I asked my wife and she offered:

Why don't we get together some time? We can do [suggested activity].