Hackman and wife Betsy Arakawa's partially mummified bodies were discovered at their New Mexico home in late February.
Authorities last week announced Hackman died at age 95 of heart disease with complications from Alzheimer's disease as much as a week after a rare, rodent-borne disease - hantavirus pulmonary syndrome - killed his 65-year-old wife.
Hackman's pacemaker last showed signs of activity on February 18, indicating an abnormal heart rhythm on the day he likely died.
The couple's bodies weren't discovered until February 26, when maintenance and security workers showed up at the Santa Fe home and alerted police, leaving a mystery for law enforcement and medical investigators to unravel.
Julia Peters, a representative for the estate of Hackman and Arakawa, urged a state district court in Santa Fe to seal records in the cases to protect the family's right to privacy in grief under the 14th Amendment to the US Constitution.
She emphasised the possibly shocking nature of photographs and video in the investigation and potential for their dissemination by media.
The request, filed on Tuesday, also described the couple's discrete lifestyle in Santa Fe since Hackman's retirement. The state capital city is known as a refuge for celebrities, artists and authors.
The couple "lived an exemplary private life for over thirty years in Santa Fe, New Mexico and did not showcase their lifestyle", said the petition.
New Mexico's open records law blocked public access to sensitive images, including depictions of people who are deceased, said Amanda Lavin, legal director at the nonprofit New Mexico Foundation for Open Government.
Some medical information also is not considered public record under the state Inspection of Public Records Act.
At the same time, the bulk of death investigations by law enforcement and autopsy reports by medical investigators were typically considered public records under state law in the spirit of ensuring government transparency and accountability, she said.
"I do think it does infringe on transparency if the court were to prohibit release of all the investigation records, including the autopsies," Lavin said on Thursday.
"The whole idea of those records being available is to ensure accountability in the way those investigations are done.
"There is also a public health concern given that hantavirus was involved."
Hackman, a Hollywood icon, won two Oscars during a storied career in films including The French Connection, Hoosiers and Superman from the 1960s until his retirement in the early 2000s.
Arakawa, born in Hawaii, studied as a concert pianist, attended the University of Southern California and met Hackman in the mid-1980s while working at a California gym.
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