For a detailed analysis of stellar chemical abundances, high-resolution spectra in the optical have mainly been used, while the development of near-infrared (NIR) spectrograph has opened new wavelength windows. Red giants have a large number of resolved absorption lines in both the optical and NIR wavelengths, but the characteristics of the lines in different wave passbands are not necessarily the same. We present a selection of FeI lines in the $z^{prime}$, $Y$, and $J$ bands (0.91-1.33 $mu$m). On the basis of two different lists of lines in this range, the Vienna Atomic Line Database (VALD) and the catalog published by Melendez & Barbuy in 1999 (MB99), we selected sufficiently strong lines that are not severely blended and compiled lists with 107 FeI lines in total (97 and 75 lines from VALD and MB99, respectively). Combining our lists with high-resolution ($lambda/Deltalambda = 28,000$) and high signal-to-noise ($>500$) spectra taken with a NIR spectrograph, WINERED, we present measurements of the iron abundances of two prototype red giants: Arcturus and $mu$ Leo. A bootstrap method for determining the microturbulence and abundance together with their errors is demonstrated. The standard deviations of $logepsilon_{rm Fe}$ values from individual FeI lines are significantly smaller when we use the lines from MB99 instead of those from VALD. With the MB99 list, we obtained $xi=1.20pm0.11 {rm km~s^{-1}}$ and $logepsilon_{rm Fe}=7.01pm0.05$ dex for Arcturus, and $xi=1.54pm0.17 {rm km~s^{-1}}$ and $logepsilon_{rm Fe}=7.73pm0.07$ dex for $mu$ Leo. These final values show better agreements with previous values in the literature than the corresponding values we obtained with VALD.
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