We have investigated the possibility of nonexplosive core contraction of massive stars in the framework of general relativity by constructing equilibrium sequences of rapidly rotating compact stars with phenomenological equations of state for the high-density region. Since there are no satisfactory equations of state for the high-temperature/high-density region, we have devised a simplified equation of state that can be considered as representing an extremely soft equation of state for the high-density region. By analyzing the stabilities of equilibrium sequences with constant total angular momentum, we have shown that there is a chance of nonexplosive core contraction in from white dwarf to neutron star density regions. It is important to note that the pressure due to a high electron fraction and/or high temperature plays an essential role in making rapidly rotating stars dynamically stable against axisymmetric collapse. In general, both a very soft equation of state and general relativity tend to make the formation of "fizzlers" more difficult. Nevertheless, since there are parameter regions that allow for fizzlers even from general relativistic computations, the possibility of their existence becomes greater than ever in spite of our oversimplified equation of state.