We present X-ray and infrared observations of the X-ray source CXOGC J174536.1-285638. Previous observations suggest that this source may be an accreting binary with a high-mass donor (HMXB) or a colliding wind binary (CWB). Based on the Chandra and XMM-Newton light curve, we have found an apparent 189 +/- 6 day periodicity with better than 99.997% confidence. We discuss several possible causes of this periodicity, including both orbital and superorbital interpretations. We explore in detail the possibility that the X-ray modulation is related to an orbital period and discuss the implications for two scenarios; one in which the variability is caused by obscuration of the X-ray source by a stellar wind, and the other in which it is caused by an eclipse of the X-ray source. We find that in the first case, CXOGC J174536.1-285638 is consistent with both CWB and HMXB interpretations, but in the second, CXOGC J174536.1-285638 is more likely a HMXB.