Abstract We simulate the evolution of binary systems with a massive primary star of 15M⊙ where we introduce an enhanced mass loss due to jets that the secondary star might launch, and find that in many cases the enhanced mass loss brings the binary system to experience the grazing envelope evolution (GEE) and form a progenitor of Type IIb supernova (SN IIb). The jets, the Roche lobe overflow (RLOF), and a final stellar wind remove most of the hydrogen-rich envelope, leaving a blue-compact SN IIb progenitor. In many cases without this jet-driven mass loss the system enters a common envelope evolution (CEE) and does not form a SN IIb progenitor. We use the stellar evolutionary code MESA binary and mimic the jet-driven mass loss with a simple prescription and some free parameters. Our results show that the jet-driven mass loss, that some systems have during the GEE, increases the parameter space for stellar binary systems to form SN IIb progenitors. We estimate that the binary evolution channel with GEE contributes about a quarter of all SNe IIb, about equal to the contribution of each of the other three channels, binary evolution without a GEE, fatal CEE (where the secondary star merges with the core of the giant primary star), and the single star channel.