Galactic nuclei are expected to be one of the main sites for formations of eccentric binary black holes (EBBHs), with an estimated detection rate of O(1−100 yr−1) with Advanced LIGO (aLIGO) detectors operating at design sensitivity. Two of the main formation channels of these binaries are gravitational capture and the secular Kozai-Lidov mechanism, with expectedly commensurable formation rates. We used Monte Carlo simulations to construct the eccentricity distributions of EBBHs formed through these channels in galactic nuclei, at the time their gravitational-wave signals enter the aLIGO band at 10 Hz. We have found that the proportion of binary black holes entering the aLIGO band with eccentricities larger than 0.1 is ∼10 percent for the secular Kozai-Lidov mechanism, and ∼90 percent for gravitational capture. We show that if future EBBH detection rates with aLIGO will be dominated by EBBHs formed in galactic nuclei, then the proportions of EBBHs formed through the two main channels can be constrained to a ΔF=0.2 wide one-sigma confidence interval with a few tens of observations, even if parameter estimation errors are taken into account at realistic levels.