We present a comprehensive study on how well gravitational-wave signals of binary black holes with nonzero eccentricities are recovered by BayesWave , a Bayesian algorithm used by the LIGO-Virgo Collaboration for unmodeled reconstructions of signal waveforms and parameters . We used two different waveform models to produce simulated signals of binary black holes with eccentric orbits and embed them in samples of simulated noise of design-sensitivity Advanced LIGO detectors . We studied the network overlaps and point estimates of central moments of signal waveforms recovered by BayesWave as a function of e , the eccentricity of the binary at 8 Hz orbital frequency . BayesWave recovers signals of near-circular ( e \lesssim 0.2 ) and highly eccentric ( e \gtrsim 0.7 ) binaries with network overlaps similar to that of circular ( e = 0 ) ones , however it produces lower network overlaps for binaries with e \in [ 0.2 , 0.7 ] . Estimation errors on central frequencies and bandwidths ( measured relative to bandwidths ) are nearly independent from e , while estimation errors on central times and durations ( measured relative to durations ) increase and decrease with e above e \gtrsim 0.5 , respectively . We also tested how BayesWave performs when reconstructions are carried out using generalized wavelets with linear frequency evolution ( chirplets ) instead of sine-Gaussian wavelets . We have found that network overlaps improve by \sim 10 - 20 percent when chirplets are used , and the improvement is the highest at low ( e < 0.5 ) eccentricities . There is however no significant change in the estimation errors of central moments when the chirplet base is used .