  We explore X-ray evidence for the presence of active galactic nuclei (AGNs)
in the two most actively star-forming Green Pea galaxies (GPs), SDSS J0749+3337
and SDSS J0822+2241, which have star-formation rates (SFRs) of $123~M_\odot$
yr$^{-1}$ and $78~M_\odot$ yr$^{-1}$, respectively. The GPs have red
mid-infrared (MIR) spectral energy distributions and higher 22 $\mu$m
luminosities than expected from a proxy of the SFR (H$\alpha$ luminosity),
consistent with hosting AGNs with 2-10 keV luminosities of $\sim10^{44}$ erg
s$^{-1}$. We thus obtain and analyze the first hard ($>$ 10 keV) X-ray data
observed with NuSTAR and archival XMM-Newton data below 10 keV. From the NuSTAR
$\approx$20 ksec data, however, we find no significant hard X-ray emission. By
contrast, soft X-ray emission with 0.5--8 keV luminosities of $\approx10^{42}$
erg s$^{-1}$ is significantly detected in both targets, which can be explained
only by star formation (SF). A possible reason for the lack of clear evidence
is that a putative AGN torus absorbs most of the X-ray emission. Applying a
smooth-density AGN torus model, we determine minimum hydrogen column densities
along the equatorial plane ($N_{\rm H}^{\rm eq}$) consistent with the
non-detection. The results indicate $N_{\rm H}^{\rm eq} \gtrsim 2\times10^{24}$
cm$^{-2}$ for SDSS J0749+3337 and $N_{\rm H}^{\rm eq} \gtrsim 5\times10^{24}$
cm$^{-2}$ for SDSS J0822+2241. Therefore, the GPs may host such heavily
obscured AGNs. Otherwise, no AGN exists and the MIR emission is ascribed to SF.
Active SF in low-mass galaxies is indeed suggested to reproduce red MIR colors.
This would imply that diagnostics based on MIR photometry data alone may
misidentify such galaxies as AGNs.
