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Magnetic skyrmions in chiral-lattice ferromagnets are currently attracting enormous research interest because of their potential applications in spintronic devices. However, they emerge in bulk specimens only in a narrow window of temperature and magnetic field. This limited stability regime is recognized as an obstacle to technical applications. Recent experiments demonstrated that the thermodynamic stability of magnetic skyrmions is enhanced or suppressed by the application of a uniaxial strain depending on its axial direction in bulk chiral-lattice ferromagnets MnSi [Y. Nii et al., Nat. Commun. 6, 8539 (2015), A. Chacon et al., Phys. Rev. Lett. 115, 267202 (2015)] and Cu2OSeO3 [S. Seki et al., Phys. Rev. B 96, 220404(R) (2017)]. Motivated by these experimental discoveries, we theoretically investigated the effects of anisotropic Dzyaloshinskii-Moriya interactions on the stability of magnetic skyrmions caused by this uniaxial strain. We find that magnetic skyrmions are significantly stabilized (destabilized) in the presence of anisotropic DM interactions when an external magnetic field lies perpendicular (parallel) to the anisotropy axis, along which the DM coupling is strengthened. Our results account completely for the experimentally observed strain-induced stabilization and destabilization of magnetic skyrmions and provide a firm ground for possible strain engineering of skyrmion-based electronic devices.PACS numbers: