In your variety of English, can "offer" be followed by a clause beginning with the word "that"?
I offered that he stay home.
or should it be
I offered that he stayed/would stay home.
In your variety of English, can "offer" be followed by a clause beginning with the word "that"?
I offered that he stay home.
or should it be
I offered that he stayed/would stay home.
I speak AmE and can think of two meanings that would work here.
First, you might mean something like the following:
My wife suggested that our son go to school today, but I offered that he stay home.
The dependent clauses use what is often called the present subjunctive; therefore, their predicates should be headed by verbs in the bare infinitive form ("go" and "stay"). However, it is a bit unusual to use the present subjunctive after the verb "offer"; verbs such as "suggest", "insist", "demand", "recommend", and "request" are more commonly used with the present subjunctive. (See here, for example.) Many people will probably find this sentence non-idiomatic for that reason.
On the other hand, you might mean something like the following:
We were trying to figure out what he had done on Monday afternoon. My wife suggested that he had gone out, but I offered that he had stayed home.
In this case, the speaker offered a possible answer to the question that they were pondering. This is correct, but I used the past perfect ("had stayed") because that seemed to make more sense with the sequence of events.
I have not addressed "would stay" because that might involve a lengthy discussion, since there are several meanings of "would". If you clarify what you mean, then I'd be happy to address whether "would stay" might be appropriate.
This is a kind of conditional statement and, in general, the simple past is accompanied by the present conditional or present continuous conditional. So the best answer is:
I offered that he would stay home.
That said, you could probably get away with the first version in spoken English, and it wouldn't be all that unusual.