Which of the following sentences is correct?
The man offers her to drink coffee.
The man offers she drinks coffee.
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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.