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Abstract Current experimental sensitivity on neutrino magnetic moments is many orders of magnitude above the Standard Model prediction. A potential measurement of next-generation experiments would therefore strongly request new physics beyond the Standard Model. However, large neutrino magnetic moments generically tend to induce large corrections to the neutrino masses and lead to fine-tuning. We show that in a model where neutrino masses are proportional to neutrino magnetic moments. We revisit, discuss and propose mechanisms that still provide theoretical consistent explanations for a potential measurement of large neutrino magnetic moments. We find only two viable mechanisms to realize large transition magnetic moments for Majorana neutrinos only.