ABSTRACT The origin of the quenching in galaxies is still highly debated. Different scenarios and processes are proposed. We use multiband (400–1600 nm) bulge–disc decompositions of massive galaxies in the redshift range 0 < z < 2 to explore the distribution and the evolution of galaxies in the $\log \, {\rm SFR-log}\: M_{*}$ plane as a function of the stellar mass weighted bulge-to-total ratio ($B/T_{M_{*}}$) and also for internal galaxy components (bulge/disc) separately. We find evidence of a clear link between the presence of a bulge and the flattening of the main sequence in the high-mass end. All bulgeless galaxies ($B/T_{M_{*}}$ < 0.2) lie on the main sequence, and there is little evidence of a quenching channel without bulge growth. Galaxies with a significant bulge component ($B/T_{M_{*}}$ > 0.2) are equally distributed in number between star forming and passive regions. The vast majority of bulges in the main-sequence galaxies are quiescent, while star formation is localized in the disc component. Our current findings underline a strong correlation between the presence of the bulge and the star formation state of the galaxy. A bulge, if present, is often quiescent, independently of the morphology or the star formation activity of the host galaxy. Additionally, if a galaxy is quiescent, with a large probability, is hosting a bulge. Conversely, if the galaxy has a discy shape is highly probable to be star forming.