We present and characterize a sample of 20 nearby Seyfert galaxies selected for having BAT 14--195keV luminosities $L_X ge 10^{41.5}$ erg s$^{-1}$, redshift $zle$0.015, being accessible for observations with the Gemini Near-Infrared Field Spectrograph (NIFS) and showing extended [OIII]$lambda$5007 emission. Our goal is to study Active Galactic Nuclei (AGN) feeding and feedback processes from near-infrared integral-field spectra, that include both ionized (HII) and hot molecular (H$_2$) emission. This sample is complemented by other 9 Seyfert galaxies previously observed with NIFS. We show that the host galaxy properties (absolute magnitudes $M_B$, $M_H$, central stellar velocity dispersion and axial ratio) show a similar distribution to those of the 69 BAT AGN. For the 20 galaxies already observed, we present surface mass density ($Sigma$) profiles for HII and H$_2$ in their inner $sim$500 pc, showing that HII emission presents a steeper radial gradient than H$_2$. This can be attributed to the different excitation mechanisms: ionization by AGN radiation for HII and heating by X-rays for H$_2$. The mean surface mass densities are in the range ($0.2le Sigma_{HII} le 35.9$)M$_odot$pc$^{-2}$, and ($0.2le Sigma_{H2} le 13.9$)$times10^{-3}$M$_odot$pc$^{-2}$, while the ratios between the HII and H$_2$ masses range between $sim$200 to 8000. The sample presented here will be used in future papers to map AGN gas excitation and kinematics, providing a census of the mass inflow and outflow rates and power as well as their relation with the AGN luminosity.
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