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Context: Methanol masers of Class I (collisionally-pumped) and Class II (radiatively-pumped) have been studied in great detail in our Galaxy in a variety of astrophysical environments such as shocks and star-forming regions and are helpful to analyze the properties of the dense interstellar medium. However, the study of methanol masers in external galaxies is still in its infancy. Aims: Our main goal is to search for methanol masers in the central molecular zone (CMZ; inner 500 pc) of the nearby starburst galaxy NGC 253. Methods: Covering a frequency range between 84 and 373 GHz ($λ$ = 3.6 to 0.8 mm) at high angular (1.6"$\sim$27 pc) and spectral ($\sim$8--9 km s$^{-1}$) resolution with the ALMA large program ALCHEMI, we have probed different regions across the CMZ of NGC 253. In order to look for methanol maser candidates, we employed the rotation diagram method and a set of radiative transfer models. Results: We detect for the first time masers above 84 GHz in NGC 253, covering an ample portion of the $J_{-1}\rightarrow(J-$ 1)$_{0}-E$ line series (at 84, 132, 229, and 278 GHz) and the $J_{0}\rightarrow(J-$ 1)$_{1}-A$ series (at 95, 146, and 198 GHz). This confirms the presence of the Class I maser line at 84 GHz, already reported but now being detected in more than one location. For the $J_{-1}\rightarrow(J-$ 1)$_{0}-E$ line series, we observe a lack of Class I maser candidates in the central star-forming disk. Conclusions: The physical conditions for maser excitation in the $J_{-1}\rightarrow(J-$ 1)$_{0}-E$ line series can be weak shocks and cloud-cloud collisions as suggested by shock tracers (SiO and HNCO) in bi-symmetric shock/active regions located in the outskirts of the CMZ. On the other hand, the presence of photodissociation regions due to a high star-formation rate would be needed to explain the lack of Class I masers in the very central regions.