The announcement from the Centers for Disease Control and Prevention represents a stepping-up of its efforts to control the outbreak.
There have been no confirmed U.S. cases of monkeypox in cats and dogs, but such pets could get the virus from close contact with an infected animal and spread it to humans, the CDC said.
"The goal is to protect people, pets and wildlife in the United States, by preventing the monkeypox virus from spreading or becoming established permanently," said Martin Cetron, deputy director of the CDC's global migration and quarantine programs.
The CDC also said laboratory tests confirmed that the monkeypox outbreak in the United States came from six African rodents that arrived on a ship from Ghana three months ago.
The CDC earlier had recommended quarantine for animals suspected of having monkeypox, but raised the order to kill those animals once it confirmed the source of the outbreak.
Monkeypox has been confirmed in 32 people, mainly in Wisconsin, Indiana and Illinois. The illness in humans is not usually fatal but causes rashes, fevers and chills. Most Americans who caught monkeypox got it from infected pet prairie dogs.
The 800 or so African rodents that arrived on the ship also should be destroyed, the CDC said. The animals have been quarantined for weeks by pet shops and other distributors.
Pet stores that comply with the new euthanasia and quarantine orders may reopen when health officials determine they are safe, the CDC said.
The government has banned imports of African rodents and halted the sale and movement of prairie dogs and some similar rodents.
People who suspect their pets have monkeypox should call state health officials, said Stephanie Shain of the Humane Society.
"If you're at all concerned, it's worth a phone call," she said. "It's difficult because the symptoms they're describing could be a number of problems — not necessarily monkeypox."