We report on the possible detection of a 55-d X-ray modulation for the ultraluminous accreting pulsar M82 X-2 from archival Chandra observations. Because M82 X-2 is known to have a 2.5-d orbital period, if the 55-d period is real, then it will be the superorbital period of the system. We also investigated variabilities of three other nearby ultraluminous X-ray sources in the central region of M82 with the Chandra data, and we did not find any evidence of periodicities. Furthermore, we re-examined the previously reported 62-d periodicity near the central region of M82 by performing a systematic timing study with all the archival Rossi X-Ray Timing Explorer and Swift data. Using various dynamic timing analysis methods, we have confirmed that the 62-d period is not stable, suggesting that it is not the orbital period of M82 X-1; this is in agreement with previous work. [ABSTRACT FROM AUTHOR]