Summary: We study, in the semiclassical limit, the singularly perturbed nonlinear Schrödinger equations \[ L^{\hslash }_{A,V} u = f(|u|^2)u \quad\text{in } \mathbb {R}^N\eqno{(0.1)} \] where \(N \geq 3\), \(L^{\hslash}_{A,V}\) is the Schrödinger operator with a magnetic field having source in a \(C^1\) vector potential \(A\) and a scalar continuous (electric) potential \(V\) defined by \[ L^{\hslash }_{A,V}= -\hslash ^2 \Delta -\frac{2\hslash }{i} A \cdot \nabla + |A|^2- \frac{\hslash }{i}\mathrm{div}A + V(x).\eqno{(0.2)} \] Here, \(f\) is a nonlinear term which satisfies the so-called Berestycki-Lions conditions. We assume that there exists a bounded domain \(\Omega \subset \mathbb {R}^N\) such that \[ m_0 \equiv \inf _{x \in \Omega } V(x) < \inf _{x \in \partial \Omega } V(x) \] and we set \(K = \{ x \in \Omega \;| \;V(x) = m_0\}\). For \(\hslash >0\) small we prove the existence of at least \({\mathrm{cupl}}(K) + 1\) geometrically distinct, complex-valued solutions to (0.1) whose moduli concentrate around \(K\) as \(\hslash \rightarrow 0\).