It is well known that a closed subspace of a normal space is normal. I am looking for a condition $*$, such that the following statement is true.
A subspace of a normal space is normal if and only if it has $*$ condition.
By a normal space, we mean a Hausdorff space that any two disjoint closed subsets contained in two disjoint open subsets.
You are looking for "completely normal space":
"A completely normal space or a hereditarily normal space is a topological space X such that every subspace of X with subspace topology is a normal space. It turns out that X is completely normal if and only if every two separated sets can be separated by neighbourhoods." - Wikipedia
and here is a proof.