Interest ( someone ) in ( something )
This phrasal verb means to persuade someone when we try offering something.
Examples :
- Can I interest you in coffee?
- Can I interest you in having a special relationship between us?
Do native speakers use this term when persuading people?
Can I also use the term as in question #2?
Solution 1:
Well according to Cambridge Dictionaries Online AmEn speakers use this term;
Can I interest you in something
› Would you like to buy or take something:
Can I interest you in a cup of coffee?
As of question # 2 I believe we do not buy or take special relationships.If you are addressing a lady I will suggest saying something like " would you allow me to persuade you to have a special relationship with me" but then native AmEn speakers know better.
Anyway, with a lady or a guy I prefer " may I" to " can I". Probably it's OK to say " may I interest you in having a special relationship with me", only our native AmEn fellow members can tell both of us.
Solution 2:
Yes people do use this phrase but it is somewhat associated with old fashioned style - especially with regard to face to face selling.
Indeed you're likely to find parodies of old style shop interactions 'Could I interest Sir in one of our fine silk ties...?" If you search Youtube.
I'm pretty sure Monty Python did several.
I would not be surprised if an older person (60 plus) used this phrase with me but someone younger would only use it in jest or parody I think.
The more common use of the verb is in the other format which you correctly distinguish from this ' I am interested in ...'