  We use GOODS and CANDELS images to identify progenitors of massive (log M >
10 Msun) compact "early-type" galaxies (ETGs) at z~1.6. Since 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~3 based on their mass, SFR and central stellar density and find that these
account for a large fraction of, and possibly all, compact ETGs at z~1.6. We
find that the average far-UV 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 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 HST images, with their stellar mass assembling in-situ, and that they have
not experienced any major merging until the epoch of observations at z~1.6.
