ABSTRACT We present an analysis of the optical-to-near-infrared (IR) counterparts of a sample of candidate dusty starbursts at z > 6. These objects were pre-selected based on the rising trend of their far-IR-to-sub-millimeter spectral energy distributions and the fact that they are radio-weak. Their precise positions are available through millimeter and/or radio interferometry, which enable us to search for their counterparts in the deep optical-to-near-IR images. The sample includes five z > 6 candidates. Three of them have their counterparts identified, one is still invisible in the deepest images, and one is a known galaxy at z = 5.667 that is completely blocked by a foreground galaxy. The three with counterparts identified are analysed using population synthesis model, and they have photometric redshift solutions ranging from 7.5 to 9.0. Assuming that they are indeed at these redshifts and that they are not gravitationally lensed, their total IR luminosities are $10^{13.8-14.1}\, {\rm L}_⊙$ and the inferred star formation rates are 6.3–13 $\times 10^3\, {\rm M}_⊙$ yr−1. The existence of dusty starbursts at such redshifts would imply that the Universe must be forming stars intensely very early in time in at least some galaxies, otherwise there would not be enough dust to produce the descendants observed at these redshifts. The inferred host galaxy stellar masses of these three objects, which are at $\gtrsim 10^{11}\, {\rm M}_⊙$ (if not affected by gravitational lensing), present a difficulty in explanation unless we are willing to accept that their progenitors either kept forming stars at a rate of $\gtrsim 10^3\, {\rm M}_⊙$ yr−1 or were formed through intense instantaneous bursts. Spectroscopic confirmation of such objects will be imperative.