To offer something, we can use various modal verbs.
I will give you a hand with that case.
Shall I give you a hand with that case?
Can I give you a hand with that case.
Will is very positive; a statement of intent more than an offer.
When we use shall as let’s the person has a chance to refuse.
Shall we go to the cinema? No, not tonight, thanks!
Below are a number of standard phrases used to make polite offers or requests:
Would you like a cup of tea?
I will see her out.
Do you happen to know the time?
Do you mind if I open the window?
Would you mind helping me?
Please don’t smoke.
I’d rather you didn’t smoke in here.
Why don’t you ask her yourself?