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.