Gene Hackman and his wife Betsy's bodies are still unclaimed -- more than a month after their tragic deaths, TMZ has learned.

TMZ reached out to the Office of the Medical Examiner in New Mexico, and they confirmed the list of unclaimed decedents is updated online every Monday. Gene and Betsy’s names appeared on the list on March 24, and they’re still there as of today.

Officials tell us it’s not unusual for bodies to remain unclaimed for a month after dying. It’s unclear why Gene and Betsy’s bodies remain unclaimed ... but it’s possible the family is still making funeral arrangements or finalizing plans.

Gene has 3 kids -- a son, Christopher, and two daughters, Leslie and Elizabeth. The sisters had spoken out in the press after the death of their dad -- who left all of his Hollywood fortune to Betsy.

Hackman had spoken about his tough relationship with his kids, admitting over the years he wasn’t around much when Chris was growing up, but he did get closer to them in his later years.

Gene and Betsy’s bodies were discovered at their Santa Fe home on February 26 ... one of their dogs was also found dead nearby.

The investigation is still ongoing, but it's been determined Betsy died on Feb. 12 from hantavirus ... while Gene died around February 17 or 18 from heart disease and advanced Alzheimer's disease.