We consider a possibility of constructing a gamma-ray source based on the multibeam configuration of a multipetawatt laser system which we simulate using a converging dipole wave. It is shown that such a configuration of fields allows generating gamma radiation with narrow directivity of about 1 mrad in the form of pulse trains or isolated pulses on the attosecond timescale. The influence of quantum electrodynamic (QED) cascade development on the parameters of generated gamma bursts is studied.