A scheme for the simulation of inviscid flows with low Mach number is derived. The scheme is built on a relaxation system and it is based on a linear implicit time discretization. The advective part is discretized by a convex combination of upwind and centered schemes, in order to recover the correct limit when the Mach number goes to zero. The implicit treatment allows to stabilize the central approximation in the low Mach limit and also to avoid demanding constraints on the time step in low Mach flows. The scheme applies to steady or unsteady flows and to general equations of state. We discuss examples pertaining to both gas and liquid flows.