Which of the following sentences is correct?
The man offers her to drink coffee.
The man offers she drinks coffee.
Which of the following sentences is correct?
The man offers her to drink coffee.
The man offers she drinks coffee.
I'm afraid neither of your options is correct! These would be options you could use instead:
The man offers her a (cup of) coffee.
The man offers to buy her a (cup of) coffee.
The man invites her to drink coffee.
The man invites her for a (cup of) coffee.
Neither sentence is good (both are ungrammatical). What I think you're trying to say is that he is offering to buy her a cup of coffee. The fact that she will drink it really has nothing to do with offering to buy it (or make it). I think you're trying to say too much in a few words. In my opinion, just leave out the drinking part entirely. It's obvious that she will drink it if she agrees to his offer.
The man offers to buy / to make her a coffee.
The man offers her a cup of coffee.