Internal magnetic states and changes in the texture of the magnetization in cobalt nanodisks have been studied using micromagnetic simulations under two different initial conditions: a nucleated Bloch type skyrmion placed at the center of the cobalt disk and displaced 32 nm from the center. Analysis of simulation results clearly demonstrates that variations in values of the anisotropy (Ku) and Dzyaloshinskii-Moriya interaction (DMI) constants have strong influence on the internal magnetic states and the energy terms of the cobalt nanodisk. Of interest is the nucleation and stability of magnetic skyrmions as a function of the DMI constant. Micromagnetic analysis suggest that the interplay between demagnetizing, exchange and Dzyaloshinskii-Moriya energies induces a diagram with different textures of the magnetization including the Bloch and Néel type skyrmions, incomplete skyrmion and skyrmionium. If the initial Bloch skyrmion state is displaced 32 nm from the center, the diagram for the texture of the magnetization changes, creating a cycloid state for higher values of the DMI constant. The results have been discussed in function of the competition between the energy terms, and the initial magnetic states.