We use a sample built on the SDSS DR7 catalogue and the bulge-disc decomposition of Simard et al. (2011, ApJS, 196, 11) to study how the bulge and disc components contribute to the parent galaxy’s star formation activity, by determining its position in the star formation rate (SFR) – stellar mass (M⋆) plane at 0.02 < z < 0.1 and around the main sequence (MS) of star-forming galaxies. For this purpose, we use the bulge and disc colours as proxy for their SFRs, while the total galaxy SFR comes from Hα or D4000. We study the mean galaxy bulge-total mass ratio (B/T) as a function of the residual from the MS (ΔMS) and find that the B/T-ΔMS relation exhibits a parabola-like shape with the peak of the MS corresponding to the lowest B/Ts at any stellar mass. The lower and upper envelope of the MS are populated by galaxies with similar B/T, velocity dispersion and concentration (R90/R50) values. The mean values of such distributions indicate that the majority of the galaxies are characterised by classical bulges and not pseudo-bulges. Bulges above the MS are characterised by blue colours or, when red, by a high level of dust obscuration, thus indicating that in both cases they are actively star forming. When on the MS or below it, bulges are mostly red and dead. At stellar masses above 1010.5M⊙, bulges on the MS or in the green valley tend to be significantly redder than their counterparts in the quiescence region, despite similar levels of dust obscuration. This could be explained with different age or metallicity content, suggesting different evolutionary paths for bulges on the MS and green valley with respect to those in the quiescence region. The disc g−r colour anti-correlates at any mass with the distance from the MS, getting redder when approaching the MS lower envelope and the quiescence region. The anti-correlation flattens as a function of the stellar mass, likely due to a higher level of dust obscuration in massive SF galaxies. We conclude that the position of a galaxy in the Log SFR – Log M⋆ plane depends on the star formation activity of its components: above the MS both bulge and disc are actively star forming. The nuclear activity is the first to be suppressed, moving the galaxies on the MS. Once the disc stops forming stars as well, the galaxy moves below the MS and eventually to the quiescence region. This is confirmed by a significant percentage (~45%) of passive galaxies with a secure two component morphology, coexisting with a population of pure spheroidals. Our findings are qualitatively in agreement with the compaction-depletion scenario, in which subsequent phases of gas inflow in the centre of a galaxy and depletion due to high star formation activity move the galaxy across the MS before the final quenching episode takes place.