Rotating proto-neutron stars can be important sources of gravitational waves to be searched for by present-day and future interferometric detectors. It was demonstrated by Imshennik that in extreme cases the rapid rotation of a collapsing stellar core may lead to fission and formation of a binary proto-neutron star which subsequently merges due to gravitational wave emission. In the present paper, we show that such dynamically unstable collapsing stellar cores may be the product of a former merger process of two stellar cores in a common envelope. We applied population synthesis calculations to assess the expected fraction of such rapidly rotating stellar cores which may lead to fission and formation of a pair of proto-neutron stars. We have used the BSE population synthesis code supplemented with a new treatment of stellar core rotation during the evolution via effective core-envelope coupling, characterized by the coupling time, $tau_c$. The validity of this approach is checked by direct MESA calculations of the evolution of a rotating 15 $M_odot$ star. From comparison of the calculated spin distribution of young neutron stars with the observed one, reported by Popov and Turolla, we infer the value $tau_c simeq 5 times 10^5$ years. We show that merging of stellar cores in common envelopes can lead to collapses with dynamically unstable proto-neutron stars, with their formation rate being $sim 0.1-1%$ of the total core collapses, depending on the common envelope efficiency.
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