The population of compact massive galaxies observed at z > 1 is hypothesized, both observationally and in simulations, to be merger remnants of gas-rich disc galaxies. To probe such a scenario, we analyse a sample of 12 gas-rich and active star-forming submillimetre galaxies (SMGs) at 1.8 < z < 3. We present a structural and size measurement analysis for all of these objects using very deep Advanced Camera for Surveys (ACS) and Near Infrared Camera and Multi-Object Spectrometer (NICMOS) imaging in the Great Observatories Origins Deep Survey (GOODS) North field. Our analysis reveals a heterogeneous mix of morphologies and sizes. We find that four galaxies (33 +/- 17 per cent) show clear signs of mergers or interactions, which we classify as early-stage mergers. The remaining galaxies are divided into two categories: five of them (42 +/- 18 per cent) are diffuse and regular disc-like objects, while three (25 +/- 14 per cent) are very compact, spheroidal systems. We argue that these three categories can be accommodated into an evolutionary sequence, showing the transformation from isolated, gas-rich discs with typical sizes of 2-3 kpc, into compact (<~1 kpc) galaxies through violent major merger events, compatible with the scenario depicted by theoretical models. Our findings that some SMGs are already dense and compact provide strong support to the idea that SMGs are the precursors of the compact, massive galaxies found at slightly lower redshift.