Wide-band Suzaku data on the merging cluster Abell 3667 were examined for hard X-ray emission in excess to the known thermal component. Suzaku detected X-ray signals in the wide energy band from 0.5 to 40 keV. The hard X-ray (> 10 keV) flux observed by the HXD around the cluster center cannot be explained by a simple extension of the thermal emission with average temperature of ~7 keV. The emission is most likely an emission from a very hot (kT > 13.2 keV) thermal component around the cluster center, produced via a strong heating process in the merger. In the north-west radio relic, no signature of non-thermal emission was observed. Using the HXD, the overall upper-limit flux within a 34x34 field-of-view around the relic is derived to be 5.3e-12 erg s-1 cm-2 in the 10-40 keV band, after subtracting the ICM contribution estimated using the XIS or the XMM-Newton spectra. Directly on the relic region, the upper limit is further tightened by the XIS data to be less than 7.3e-13 erg s-1 cm-2, when converted into the 10--40 keV band. The latter value suggest that the average magnetic field within the relic is higher than 1.6 uG. The non-thermal pressure due to magnetic fields and relativistic electrons may be as large as ~20% of the thermal pressure in the region.