United Airlines has accepted "full responsibility" for a dog's in-flight death after a flight attendant put the pet in the overhead locker.

"This was a tragic accident that should never have happened," the airline said.

The French bulldog died during a flight from Houston to New York on Monday.

Witnesses said the flight attendant had asked one of the passengers to put her airline-approved pet carrier in the locker. The attendant later said she did not know the dog was in the bag.

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"We assume full responsibility for this tragedy and express our deepest condolences to the family and are committed to supporting them," the US airline said in a statement.

"We are thoroughly investigating what occurred to prevent this from ever happening again. Pets should never be placed in the overhead bin," the statement added.

Passenger Maggie Gremminger, who said she was seating behind the woman with the dog, gave her account of what happened.

"I witnessed a United flight attendant instruct a woman to put her dog carrier with live dog in an overhead bin," she told One Mile at a Time, a travel website.

"The passenger adamantly pushed back, sharing verbally that her dog was in the bag.

"The flight attendant continued to ask the passenger to do it, and she eventually complied.

"By the end of the flight, the dog was dead. The woman was crying in the airplane aisle on the floor."

The eyewitness went on Twitter, saying that "my heart is broken".

Although overhead lockers are not air-tight, lack of oxygen could have been the cause of the dog's death, US media report.

I want to help this woman and her daughter. They lost their dog because of an @united flight attendant. My heart is broken. pic.twitter.com/mjXYAhxsAq

— MaggieGremminger (@MaggieGrem) March 13, 2018

United's policy for onboard animals states that "a pet travelling in cabin must be carried in an approved hard-sided or soft-sided kennel.

"The kennel must fit completely under the seat in front of the customer and remain there at all times."