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Determining the effective temperatures (T eff) of red supergiants (RSGs) observationally is important in many fields of stellar physics and galactic astronomy, yet some significant difficulties remain as a result of the model uncertainty, which originates mainly in the extended atmospheres of RSGs. Here we propose a line-depth ratio (LDR) method in which we use only Fe  i lines. As opposed to the conventional LDR method, in which lines of multiple species are involved, this LDR method is insensitive to surface gravity effects and is expected to circumvent the uncertainty originating in the upper atmosphere of RSGs. Therefore, the LDR– T eff relationships that we calibrated empirically with red giants can be directly applied to RSGs, although various differences between the two groups of objects, for example caused by 3D non-local thermodynamic equilibrium (non-LTE) effects, need to be kept in mind. Using the near-infrared YJ -band spectra of nine well-known solar-metallicity red giants observed with the WINERED high-resolution spectrograph, we selected 12 pairs of Fe  i lines that were the least contaminated with other lines. After applying their LDR– T eff relation to 10 nearby RSGs, the resulting T eff with an internal precision of |$30\!-\!70\, \mathrm{K}$| shows good agreement with previous observational results assuming 1D LTE and with the Geneva stellar evolution model. We found no evidence of significant systematic bias caused by various differences, including those in the size of the non-LTE effects, between red giants and RSGs, except for one line pair that we rejected because the non-LTE effects may be as large as |$\sim \! 250\, \mathrm{K}$|⁠. Nevertheless, it is difficult to evaluate the systematic bias, and further study is required, for example including the 3D non-LTE calculations of all the lines involved. [ABSTRACT FROM AUTHOR]