Metallicity is known to significantly affect the radial expansion of a massive star: the lower the metallicity the more compact the star, especially during its post-MS evolution. Our goal is to study the impact of this effect in the context of binary evolution. Using the stellar-evolution code MESA, we compute evolutionary tracks of massive stars at six different metallicities between 1.0 Z and 0.01 Z . We explore variations of factors known to affect the radial expansion of massive stars (eg. semiconvection, overshooting, rotation). Using observational constraints, we find support for evolution in which already at metallicity Z ≈ 0.2 Z massive stars stay relatively compact (∼ 100 R ) during the Hertzprung-Gap phase (HG) and most of their expansion happens during core-helium burning (CHeB). Consequently, we show that metallicity has a strong influence on the type of mass transfer evolution in binary systems. At solar metallicity a case-B mass transfer is initiated shortly after the end of MS and a giant donor is almost always a rapidly-expanding HG star. However, at lower metallicity the parameter space for mass transfer from a more evolved, slowly-expanding CHeB star increases dramatically. This means that envelope stripping and formation of helium stars in low metallicity environments happens later during the evolution of the donor, implying a shorter duration of the Wolf-Rayet phase (even by an order of magnitude) and higher final core masses. This metallicity effect is independent of the impact of metallicity-dependent stellar winds. At metallicities Z ≤ 0.04 Z a significant fraction of massive stars in binaries with periods above 100 days engage in their first episode of mass transfer very late into their evolution, when they already have a well developed CO core. The remaining lifetime ( 10 4 yr) is unlikely to be enough to strip the entire H-rich envelope. Cases of unstable mass transfer leading to a merger would produce CO cores that are fast spinning at the moment of collapse. We find that the parameter space for mass transfer from massive donors (> 40 M ) with outer convective envelopes is extremely small or even non-existent. We briefly discuss this finding in the context of formation of binary black hole mergers.