  The general $d$-position number ${\rm gp}_d(G)$ of a graph $G$ is the
cardinality of a largest set $S$ for which no three distinct vertices from $S$
lie on a common geodesic of length at most $d$. This new graph parameter
generalizes the well studied general position number. We first give some
results concerning the monotonic behavior of ${\rm gp}_d(G)$ with respect to
the suitable values of $d$. We show that the decision problem concerning
finding ${\rm gp}_d(G)$ is NP-complete for any value of $d$. The value of ${\rm
gp}_d(G)$ when $G$ is a path or a cycle is computed and a structural
characterization of general $d$-position sets is shown. Moreover, we present
some relationships with other topics including strong resolving graphs and
dissociation sets. We finish our exposition by proving that ${\rm gp}_d(G)$ is
infinite whenever $G$ is an infinite graph and $d$ is a finite integer.
