Concept:
In reported speech, if the reporting verb is in the past tense, then the tense of the reported clause usually changes.
Here the reporting verb is:
\[
\text{told}
\]
which is past tense.
Step 1: Identify the incorrect part.
The sentence uses:
\[
\text{he will}
\]
But since the reporting verb is past tense, \(will\) changes to
\[
would
\]
Step 2: Correct the sentence.
\[
\text{My close friend told me that he would tell a very important information.}
\]
According to the options, the correct replacement is:
\[
\text{he would}
\]
Step 3: Final answer.
\[
\boxed{\text{he would}}
\]