We use GOODS and CANDELS images to identify progenitors of massive (M > 10(10) M-circle dot) compact early-type galaxies (ETGs) at z similar to 1.6. Because merging and accretion increase the size of the stellar component of galaxies, if the progenitors are among known star-forming galaxies, these must be compact themselves. We select candidate progenitors among compact Lyman-break galaxies at z similar to 3 on the basis of their mass, star-formation rate (SFR), and central stellar density, and we find that these account for a large fraction of, and possibly all, compact ETGs at z similar to 1.6. We find that the average far-UV spectral energy distribution (SED) of the candidates is redder than that of the non-candidates, but the optical and mid-IR SED are the same, implying that the redder UV of the candidates is inconsistent with larger dust obscuration and consistent with more evolved (aging) star formation. This is in line with other evidence suggesting that compactness is a sensitive predictor of passivity among high-redshift massive galaxies. We also find that the light distribution of both the compact ETGs and their candidate progenitors does not show any extended "halos" surrounding the compact "core," both in individual images and in stacks. We argue that this is generally inconsistent with the morphology of merger remnants, even if gas rich, as predicted by N-body simulations. This suggests that the compact ETGs formed via highly dissipative, mostly gaseous accretion of units whose stellar components are very small and undetected in the Hubble Space Telescope images, with their stellar mass assembling in situ, and that they have not experienced any major merging until the epoch of observations at z similar to 1.6.