Several Type IIb supernovae (SNe IIb) have been extensively studied, both in terms of the progenitor radius and the mass-loss rate in the final centuries before the explosion. While the sample is still limited, evidence has been accumulating that the final mass-loss rate tends to be larger for a more extended progenitor, with the difference exceeding an order of magnitude between the more and less extended progenitors. The high mass-loss rates inferred for the more extended progenitors are not readily explained by a prescription commonly used for a single stellar wind. In this paper, we calculate a grid of binary evolution models. We show that the observational relation in the progenitor radii and mass-loss rates may be a consequence of non-conservative mass transfer in the final phase of progenitor evolution without fine tuning. Further, we find a possible link between SNe IIb and SNe IIn. The binary scenario for SNe IIb inevitably leads to a population of SN progenitors surrounded by dense circumstellar matter (CSM) due to extensive mass loss (Ṁ 10 −4 M ⊙ yr −1 ) in the binary origin. About 4 % of all observed SNe IIn are predicted to have dense CSM, produced by binary non-conservative mass transfer, whose observed characteristics are distinguishable from SNe IIn from other scenarios. Indeed, such SNe may be observationally dominated by systems experiencing huge mass loss in the final 10 3 yr, leading to luminous SNe IIn or initially bright SNe IIP or IIL with a characteristics of SNe IIn in their early spectra.