Hackman, 95, was found dead on Wednesday in a mudroom and his 65-year-old wife, Betsy Arakawa, was found in a bathroom next to a space heater, Santa Fe County Sheriff's Office detectives wrote in a search warrant.
There was an open prescription bottle and pills scattered on a countertop near Arakawa.
Denise Avila, a sheriff's office spokesperson, said there was no indication they had been shot or had any wounds.
The New Mexico Gas Co tested the gas lines in and around the home after the bodies were discovered, according to the warrant.
At the time, it didn't find any signs of problems and the Fire Department found no signs of a carbon monoxide leak or poisoning.
A sheriff's detective wrote there were no obvious signs of a gas leak, but he noted people exposed to gas leaks or carbon monoxide might not show signs of poisoning.
The gruff-but-beloved Hackman was among the best actors of his generation, appearing as villains, heroes and antiheroes in dozens of dramas, comedies and action films from the 1960s until his retirement in the early 2000s.
"He was loved and admired by millions around the world for his brilliant acting career, but to us he was always just Dad and Grandpa. We will miss him sorely and are devastated by the loss," his daughters and granddaughter said in a statement Thursday.
A maintenance worker reported the home's front door was open when he arrived to do routine work on Wednesday, and he called police after finding the bodies, investigators said.
He and another worker said they rarely saw the homeowners and their last contact with them had been about two weeks earlier.
Hackman appeared to have fallen, a deputy observed. He was wearing a blue t-shirt, gray sweatpants and slippers. A pair of sunglasses and a cane were nearby.
A dead German shepherd was found in a bathroom closet near Arakawa, police said. Two healthy dogs were found on the property - one inside and one outside.
Hackman routinely showed up on Hollywood list of greatest American actors of the 20th century. He could play virtually any kind of role, from comic book villain Lex Luthor in Superman to a coach finding redemption in the sentimental favourite Hoosiers.
Hackman was a five-time Oscar nominee who won for The French Connection in 1972 and Unforgiven two decades later. His death comes just four days before this year's ceremony.
Tributes quickly poured in from Hollywood.Â
"There was no finer actor than Gene. Intense and instinctive. Never a false note. He was also a dear friend whom I will miss very much," actor-director Clint Eastwood, Hackman's Unforgiven co-star, said in a statement.
Hackman met Arakawa, a classically trained pianist who grew up in Hawaii, when she was working part-time at a California gym in the mid-1980s, the New York Times reported in 1989. They soon moved in together, and by the end of the decade relocated to Santa Fe.
Their Southwestern-style ranch on Old Sunset Trail sits on a hill in a gated community with views of the Rocky Mountains.
The sprawling four-bedroom home on six acres was built in 1997 and had an estimated market value of a little over $US4 million ($A6.4 million), according to Santa Fe County property tax records.
Hackman and his wife also owned a more modest home next door.
Hackman also co-wrote three novels, starting with the swashbuckler, Wake of the Perdido Star, with Daniel Lenihan in 1999, according to publisher Simon & Schuster.
He then penned two by himself, concluding with Pursuit in 2013, about a female police officer on the tail of a predator.
In his first couple decades in New Mexico, Hackman was often seen around the historic state capital, known as an artist enclave, tourism destination and retreat for celebrities.
In recent years, he was far less visible, though even the most mundane outings caught the attention of the press.
Aside from appearances at awards shows, he was rarely seen in the Hollywood social circuit and retired from acting about 20 years ago. His was the rare Hollywood retirement that actually lasted.
Hackman had three children from a previous marriage. He and Arakawa had no children together but were known for having German shepherds.
