Flexible structures have recently been considered as alternative ways to extract energy from ocean waves (Alam 2012, Desmars et al 2018) or tidal currents (Trasch et al 2018), with the objective to find devices with complementary working characteristics compared to non-flexible energy harvesters. We investigate the dynamics of a finite length tensioned membrane with a localized linear damper to mimic energy extraction, which is placed in a uniform current parallel to a free surface. Such configuration resembles the so called Nemtsov’s membrane (Nemtsov 1985), recently studied in details and generalized to finite depth cases by Labarbe & Kirillov (2020, 2021), or to the infinite flag configuration close to a free surface studied recently by Mougel & Michelin (2020). The above studies reveal the importance of the free surface on the stability of the system, due to interactions between surface waves and structural waves when a current is present. In the present study, focus is placed on both forcing by incident waves (as already reported by Achour et al 2020 for weak currents) and stability analysis in order to investigate the role of the current on wave energy extraction by a flexible membrane, and shed additional light on the possible instability mechanism. In this objective, a linear potential flow model coupled to a tensioned beam is considered, and numerical results computed with the finite elements code FreeFEM++ (Hecht 2012) interfaced by StabFEM solver (Fabre et al 2019) are presented for a large range of physical parameters covering both subcritical and supercritical regimes.