1. I said to her, "Will you close the door?" -direct narration-
2. I told her that she would close the door. -indirect narration-
3. I asked/begged her to close the door? -indirect narration-
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#1 can be changed into #3. What about #2? Can #1 be changed into #2?
1 and 3 are closest in meaning.
1 and 2 don't mean the same thing. Sentence 2 is written as a prediction or a command!