For a given fixed poset $P$ we say that a family of subsets of $[n]$ is $P$-saturated if it does not contain an induced copy of $P$, but whenever we add to it a new set, an induced copy of $P$ is formed. The size of the smallest such family is denoted by $sat\ast (n,P)$. For the diamond poset $D2$ (the two-dimensional Boolean lattice), Martin, Smith and Walker proved that $n\le sat\ast (n,D2)\le n+1$. In this paper we prove that $sat\ast (n,D2)\ge (22-o(1))n$. We also explore the properties that a diamond-saturated family of size $cn$, for a constant $c$, would have to have.