The strong dependence of the neutrino annihilation mechanism on the mass accretion rate makes it difficult to explain the LGRBs with duration in excess of 100 seconds as well as the precursors separated from the main gamma-ray pulse by few hundreds of seconds. Even more difficult is to explain the Swift observations of the shallow decay phase and X-ray flares, if they indeed indicate activity of the central engine for as long as 10^4 seconds. These data suggest that some other, most likely magnetic mechanisms have to be considered. Since the efficiency of magnetic mechanisms does not depend that much on the mass accretion rate, the magnetic models do not require the development of accretion disk within the first few seconds of the stellar collapse and hence do not require very rapidly rotating stellar cores at the presupernova state. This widens the range of potential LGRB progenitors. In this paper, we re-examine the close binary scenario allowing for the possibility of late development of accretion disks in the collapsar model and investigate the available range of mass accretion rates, black hole masses, and spins. We find that the black hole mass can be much higher than 2-3 Msun, usually assumed in the collapsar model, and normally exceeds half of the presupernova mass. The black hole spin is rather moderate, a=0.4-0.8, but still high enough for the Blandford-Znajek mechanism to remain efficient provided the magnetic field is sufficiently strong. Our numerical simulations confirm the possibility of magnetically driven stellar explosions, in agreement with previous studies, but point towards the required magnetic flux on the black hole horizon in excess of 10^28 G cm^2. At present, we cannot answer with cirtainty whether such a strong magnetic field can be generated in the stellar interior. Perhaps, the supernova explosions associated with LGRBs are still neutrino-driven and their gamma-ray signature is the precursors. The supernova blast cleares up escape channels for the magnetically driven GRB jets, which may produce the main pulse. In this scenario, the requirements on the magnetic field strength can be lowered. A particularly interesting version of the binary progenitor involves merger of a WR star with an ultra-compact companion, neutron star or black hole. In this case we expect the formation of very long-lived accretion disks, that may explain the phase of shallow decay and X-ray flares observed by Swift. Similarly long-lived magnetic central engines are expected in the current single star models of LGRB progenitors due to their assumed exceptionally fast rotation.