We investigate how the stellar and gas-phase He abundances evolve as a function of time within simulated star-forming disc galaxies with different star formation histories. We make use of a cosmological chemodynamical simulation for galaxy formation and evolution, which includes star formation as well as energy and chemical enrichment feedback from asymptotic giant branch stars, core-collapse supernovae, and Type Ia supernovae. The predicted relations between the He mass fraction, Y, and the metallicity, Z, in the interstellar medium of our simulated disc galaxies depend on the galaxy star formation history. In particular, dY/dZ is not constant and evolves as a function of time, depending on the specific chemical element that we choose to trace Z; in particular, dY/dX_O and dY/dX_C increase as a function of time, whereas dY/dX_N decreases. In the gas-phase, we find negative radial gradients of Y, due to the inside-out growth of our simulated galaxy discs as a function of time; this gives rise to longer chemical enrichment timescales in the outer galaxy regions, where we find lower average values for Y and Z. Finally, by means of chemical-evolution models, in the galactic bulge and inner disc, we predict steeper Y vs. age relations at high Z than in the outer galaxy regions. We conclude that for calibrating the assumed Y − Z relation in stellar models, C, N, and C+N are better proxies for the metallicity than O because they show steeper and less scattered relations.