We demonstrate that $∼10$ seconds after core-collapse of a massive star, a thermonuclear explosion of the outer shells is possible for some (tuned) initial density and composition profiles, assuming the neutrinos failed to explode the star. The explosion may lead to a successful supernova, as first suggested by Burbidge, Burbidge, Fowler and Hoyle (1957). We perform a series of one-dimensional (1D) calculations of collapsing massive stars with simplified initial density profiles (similar to the results of stellar evolution calculations) and various compositions (not similar to 1D stellar evolution calculations). We assume that the neutrinos escaped with negligible effect on the outer layers, which inevitably collapse. As the shells collapse, they compress and heat up adiabatically, enhancing the rate of thermonuclear burning. In some cases, where significant shells of mixed helium and oxygen are present with pre-collapsed burning times of $\lesssim100\,\textrm{s}$ ($≈10$ times the free-fall time), a thermonuclear detonation wave is ignited, that unbinds the outer layers of the star, leading to a supernova. The energy released is small $\lesssim10^{50}\,\textrm{erg}$ and negligible amounts of synthesized material (including $^{56}$Ni) are ejected, implying that these 1D simulations are unlikely to represent typical core-collapse supernovae. However, they serve as a proof of concept that the core-collapse induced thermonuclear explosions are possible. More realistic two-dimensional and three-dimensional simulations, which are possible with current computational capabilities, are required to study this mechanism as a possible primary channel for core-collapse supernovae. Preliminary two-dimensional calculations that include rotation indicate that stronger explosions are possible. ; Comment: 12 pages, 8 figures