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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.

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    One is that any two closed disjoint closed subsets of $X$ have disjoint closures in some compactification of $X.$ – 2017-01-18
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    Is there any reference for your comment? – 2017-01-18
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    Probably in General Topology by Engelking. Chapter 3.Section 3.5: Compactifications. You may also want to see Theorem 2.1.7 (Chapter 2, Section 1: Subpspaces) on hereditary normality. – 2017-01-18

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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.