The role of cubic ice, ice Ic, in the nucleation of ice from supercooled water has been widely debated in the past decade. Computer simulations can provide insightful information about the mechanism of ice nucleation at a molecular scale. In this work, we use molecular dynamics to study the competition between ice Ic and hexagonal ice, ice Ih, in the process of ice nucleation. Using a seeding approach, in which classical nucleation theory is combined with simulations of ice clusters embedded in supercooled water, we estimate the nucleation rate of ice for a pathway in which the critical nucleus has an Ic structure. Comparing our results with those previously obtained for ice Ih [Sanz et al., J. Am. Chem. Soc. 135, 15008 (2013)], we conclude that within the accuracy of our calculations both nucleation pathways have the same rate for the studied water models (TIP4P/Ice and TIP4P/2005). We examine in detail the factors that contribute to the nucleation rate and find that the chemical potential difference with the fluid, the attachment rate of particles to the cluster, and the ice-water interfacial free energy are the same within the estimated margin of error for both ice polymorphs. Furthermore, we study the morphology of the ice clusters and conclude that they have a spherical shape.