What topological spaces satisfy a property involving relatively compact sets?

A subset of a topological space is called relatively compact if its closure is compact. My question is, what kind of topological spaces satisfy the following property: for every relatively compact set $$S$$, there exists a relatively compact set $$T$$ such that the closure of $$S$$ is a subset of the interior of $$T$$?

Is there some category of topological spaces which satisfies this property? And is there an example of a $$T_1$$ space which does not satisfy this property?

My reason for asking this, by the way, is that relatively compact subsets of a $$T_1$$ form a bornology, and this property says that that bornology interacts well with the topology.

If a topological space satisfies your property, then a singleton point set $$\{x\}$$ is compact, and hence there must be a relatively compact set $$T$$ whose interior contains $$x$$ (in fact, it contains $$\overline{\{x\}}$$, which may be strictly larger). The closure of $$T$$ is a compact neighbourhood of $$x$$.
If, on the other hand, if we have locally compact space and a relatively compact set $$S$$, then $$\overline{S}$$ is compact. For each point $$s \in \overline{S}$$, there is an open neighbourhood $$\mathcal{N}_s$$ of $$s$$ that is compact. Note that the interiors of these neighbourhoods cover $$\overline{S}$$, hence there must be a finite-subcover of $$\overline{S}$$. Unioning this finite subcover yields an open, relatively compact neighbourhood containing $$\overline{S}$$, establishing equivalence.