  Given a graph $G$, a set $X$ of vertices in $G$ satisfying that between every
two vertices in $X$ (respectively, in $G$) there is a shortest path whose
internal vertices are not in $X$ is a mutual-visibility (respectively, total
mutual-visibility) set in $G$. The cardinality of a largest (total)
mutual-visibility set in $G$ is known under the name (total) mutual-visibility
number, and has been studied in several recent works.
  In this paper, we propose two lower variants of the mentioned concepts,
defined as the smallest possible cardinality among all maximal (total)
mutual-visibility sets in $G$, and denote them by $\mu^{-}(G)$ and
$\mu_t^{-}(G)$, respectively. While the total mutual-visibility number is never
larger than the mutual-visibility number in a graph $G$, we prove that both
differences $\mu^{-}(G)-\mu_t^{-}(G)$ and $\mu_t^{-}(G)-\mu^{-}(G)$ can be
arbitrarily large. We characterize graphs $G$ with some small values of
$\mu^{-}(G)$ and $\mu_t^{-}(G)$, and prove a useful tool called Neighborhood
Lemma, which enables us to find upper bounds on the lower mutual-visibility
number in several classes of graphs. We compare the lower mutual-visibility
number with the lower general position number, and find a close relationship
with Bollob\'{a}s-Wessel theorem when this number is considered in Cartesian
products of complete graphs. Finally, we also prove the NP-completeness of the
decision problem related to $\mu_t^{-}(G)$.
