I'm not sure about the use of the verbs "offer" and "promise" with to infinitive clauses. With the verb offer, the noun following the verb (person offered something) is normally understood as the subject of the to-infinitive like:
We offered her to stay with us until she found an apartment.
(we offered, she to stay)
But, can the context be ambiguous and allow interpreting the subject of the main clause as the subject of to-infinitive clause, like:
We offered her to pay the rent until she found an apartment.
She to pay the rent or we to pay the rent?
My second question is related to the same construction with the verb "promise". Am I correct that "that" clause phrasing is usually preferred to the one with "to-infinitive" with this verb, especially with negative sentences:
I promised her not to forget to call her when I came home.
vs
I promised her that I wouldn't forget to call her when I came home.
Best Answer
"We offered her to stay with us" is not standard English. With the verb "to offer", the infinitive must come immediately after the main verb, as in "We offered to let her stay with us." Similarly, you cannot say "We offered her to pay the rent." You have to say something like "We offered to pay the rent for her."
The verb "to promise" behaves differently, and either of the examples you give is correct. I would use the first version myself, but it's purely a matter of style.