In light of the newly opened and rapidly growing GW window in multi-messenger astronomy, in order to fully take advantage of the new opportunities we are provided with, new ideas are required for a better and deeper employ of the state-of-the-art probes we handle. Following this goal, here we suggest a method to constrain the cosmological background, and the Hubble constant in particular, by future observations of gravitationally lensed radiation emitted by a single source in both the gravitational wave and the electromagnetic regimes. The lensing of the gravitational wave radiation, in fact, can leave a clear imprinting in the corresponding waveform, and we want to analyze if such kind of measurements can be successfully employed to better constrain the cosmological background. Thus, by making use of wave optics for the gravitational wave lensed signal, and of standard geometrical optics approximation for the electromagnetic one, we study the impact of different cosmological parameters on the value of the arrival time delay due to gravitational lensing, given specific GW frequencies, mass models of the lens, and redshifts and positions (with respect to the lens) of the source. Although the rate of lensing of gravitational waves is expected to be low, we show that even one single event could provide us with an uncertainty on $H_0$ comparable with present independent probes in a "pessimistic" scenario (with a pulsar population similar to present Pulsar Timing Array state), and of two orders smaller in an optimistic one (with a number of observed pulsars as large as that expected from the Square Kilometer Array). Thus, its role in the solution of the Hubble tension could be decisive.