Whici of the following sentence is correct? "I have talked with him today and he said that he is well"
"I talked with him today and he said that he is well"
Which tense I need to use at the sentence? Present Perfect or Past tense?
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They're both correct, in this regard.
The past and present perfect have overlapping uses, and this is in the overlap.
Because you are suggesting that you assume his health continues to be well to the current time (through your use of the present tense, rather than choosing to backshift the reported speech), I'd favour toward the present perfect with its use to speak of things that continue to influence the present (or close to the present).
However, while the simple past does not in itself suggest this, the sentence as a whole is still a correct sentence that expresses the same information.