Formation of amyloid-like protein aggregates in human organs and tissues underlies many serious diseases, therefore being in the focus of numerous biochemical, medical, and molecular biological studies. So far, formation of amyloids by globular proteins has been studied mostly under conditions that strongly destabilized their native structure. Here we present our results obtained at permissive temperature by thioflavin T fluorescence, far UV CD, IR spectroscopy, and electron microscopy. We used apomyoglobin and its mutants with Ala or Phe substituted for Val10 that are structurally close to wild type apomyoglobin. It is shown that at permissive temperature the ability of the protein to form amyloids depends on the extent of its structural destabilization, but not on hydrophobicity of the substituting residue. A possible difference between amyloids formed by strongly destabilized proteins and those yielded by proteins with a slightly fluctuating native structure, as well as the stroke and infarction effect on the ability of proteins to form amyloid structures, are discussed.