[en] The effects of external (hydrostatic) compression and internal (chemical) expansion on the skyrmion phase have been investigated in Cu_2OSeO_3 single crystal and Cu_2OSe_1_−_xTe_xO_3 (0  ⩽  x  ⩽  0.2) polycrystalline samples respectively. Under external pressure, the ferrimagnetic temperature T _C ∼ 57 K is increased with a positive rate (dT _C /dP  =  0.27 K kbar"−"1). The skyrmion zone enclosed in the H-T phase diagram is enlarged and shifted to high temperatures for Cu_2OSeO_3 single crystal. Consistently, the T _C decreases with a negative rate (dT _C /dx  =  −15.5 K/x) in Cu_2OSe_1_−_xTe_xO_3 (0  ⩽  x  ⩽  0.2) and the skyrmion zone is suppressed and shifted to the low temperature side; complete disappearance of the skyrmion phase has been noted for x  =  0.2. These observed results are explained to be related to the local crystal structure, namely, inter-atomic bond lengths and bond angles that modulate the strengths of competing Heisenberg exchange and Dzyaloshinsky–Moriya (DM) interactions. This approach to tuning the skyrmion ground state by external and chemical pressures might be useful for designing of spintronics devices. (paper)