  We introduce a notion of approximate orthogonality preserving mappings
between Hilbert $C^*$-modules. We define the concept of $(\delta,
\varepsilon)$-orthogonality preserving mapping and give some sufficient
conditions for a linear mapping to be $(\delta, \varepsilon)$-orthogonality
preserving. In particular, if $\mathscr{E}$ is a full Hilbert
$\mathscr{A}$-module with $\mathbb{K}(\mathscr{H})\subseteq \mathscr{A}
\subseteq \mathbb{B}(\mathscr{H})$ and $T, S:\mathscr{E}\longrightarrow
\mathscr{E}$ are two linear mappings satisfying $\big|\langle Sx,
Sy\rangle\big| = \|S\|^2\,|\langle x, y\rangle|$ for all $x, y\in \mathscr{E}$
and $\|T - S\| \leq \theta \|S\|$, then we show that $T$ is a $(\delta,
\varepsilon)$-orthogonality preserving mapping. We also prove whenever
$\mathbb{K}(\mathscr{H})\subseteq \mathscr{A} \subseteq
\mathbb{B}(\mathscr{H})$ and $T: \mathscr{E} \longrightarrow \mathscr{F}$ is a
nonzero $\mathscr{A}$-linear $(\delta, \varepsilon)$-orthogonality preserving
mapping between $\mathscr{A}$-modules, then $$\big\|\langle Tx, Ty\rangle -
\|T\|^2\langle x, y\rangle\big\|\leq \frac{4(\varepsilon - \delta)}{(1 -
\delta)(1 + \varepsilon)} \|Tx\|\,\|Ty\|\qquad (x, y\in \mathscr{E}).$$ As a
result, we present some characterizations of the orthogonality preserving
mappings.
