We consider the standard model of first-passage percolation on $\mathbb{Z}^d$ ($d\geq 2$), with i.i.d. passage times associated with either the edges or the vertices of the graph. We focus on the particular case where the distribution of the passage times is the Bernoulli distribution with parameter $1-ε$. These passage times induce a random pseudo-metric $T_ε$ on $\mathbb{R}^d$. By subadditive arguments, it is well known that for any $z\in\mathbb{R}^d\setminus \{0\}$, the sequence $T_ε(0,\lfloor nz \rfloor) / n$ converges a.s. towards a constant $μ_ε(z)$ called the time constant. We investigate the behavior of $ε\mapsto μ_ε(z)$ near $0$, and prove that $μ_ε(z) = \| z\|_1 - C (z) ε^{1/d_1(z)} + o ( ε^{1/d_1(z)}) $, where $d_1(z)$ is the number of non null coordinates of $z$, and $C(z)$ is a constant whose dependence on $z$ is partially explicit.