In order to understand the nature of the sources producing the recently uncovered CIB fluctuations , we study cross-correlations between the fluctuations in the source-subtracted Cosmic Infrared Background ( CIB ) from Spitzer /IRAC data and the unresolved Cosmic X-ray Background ( CXB ) from deep Chandra observations . Our study uses data from the EGS/AEGIS field , where both datasets cover an \simeq 8 ^ { \prime } \times 45 ^ { \prime } region of the sky . Our measurement is the cross-power spectrum between the IR and X-ray data . The cross-power signal between the IRAC maps at 3.6 \mu m and 4.5 \mu m and the Chandra [ 0.5-2 ] keV data has been detected , at angular scales { { } _ { > } \atop { } ^ { \sim } } 20 { { } ^ { \prime \prime } } , with an overall significance of \simeq 3.8 \sigma and \simeq 5.6 \sigma , respectively . At the same time we find no evidence of significant cross-correlations at the harder Chandra bands . The cross-correlation signal is produced by individual IR sources with 3.6 \mu m and 4.5 \mu m magnitudes m _ { AB } { { } _ { > } \atop { } ^ { \sim } } 25-26 and [ 0.5-2 ] keV X-ray fluxes \ll 7 \times 10 ^ { -17 } erg / cm ^ { 2 } / s . We determine that at least 15 - 25 \% of the large scale power of the CIB fluctuations is correlated with the spatial power spectrum of the X-ray fluctuations . If this correlation is attributed to emission from accretion processes at both IR and X-ray wavelengths , this implies a much higher fraction of accreting black holes than among the known populations . We discuss the various possible origins for the cross-power signal and show that neither local foregrounds , nor the known remaining normal galaxies and active galactic nuclei ( AGN ) can reproduce the measurements . These observational results are an important new constraint on theoretical modeling of the near-IR CIB fluctuations .