The coronavirus epidemic is exacerbating a meat glut in the U.S., filling cold-storage facilities with pork, chicken and beef intended for export to markets hit by the outbreak.

Rising quantities of breasts, thigh meat and drumsticks lifted the amount of chicken in U.S. cold storage facilities by 12% over the first month of the year to 957.5 million pounds, the highest level on record for the month of January, according to the U.S. Department of Agriculture. The amount of pork in storage climbed 11% versus January 2019. Meat industry executives say those stockpiles have likely since grown.

“The cold storages we deal with are all busting at the seams,” Joe Sanderson, chief executive of Mississippi-based chicken company Sanderson Farms Inc., said on a conference call last week.

U.S. meatpackers including Sanderson, Tyson Foods Inc. TSN 2.76% and Pilgrim’s Pride Corp. PPC 2.99% have been counting on big orders from Chinese buyers, as trade tensions ease and China struggles to fill a pork shortfall after losing hundreds of millions of hogs to a swine disease. U.S. hog farmers have ramped up production, and chicken companies have been raising fatter birds. U.S. meat exports to China surged in late 2019, with more U.S. pork shipped to the country in November and December than in all of 2018, according to the USDA.

The new coronavirus has scrambled that outlook. Huge numbers of people staying home rather than eating out slowed meat consumption in China, which hurt orders from the U.S. Government-mandated quarantines in China created logistical snarls.

“The ports are basically backed up,” Tyson’s CEO Noel Whitesaid at an investor event last month. Four China-bound ships loaded with Sanderson chicken were diverted from their destinations in recent weeks, although eventually all of them landed, the company’s executives said.

Cold storage warehouses serve as staging grounds where vast quantities of meat, along with french fries, seafood and other products, are held before delivery to domestic buyers or shipping ports. Meat can be kept frozen in cold storage warehouses for months, at a cost to the companies that produced it.

Lineage Logistics Holding LLC’s cold storage warehouses in shipping hubs like Los Angeles and Seattle are close to full, said Brian Beattie, senior vice president of sales. Pallets of shrink-wrapped boxed pork, chicken and beef are stacked on racks that rise nearly 60 feet to the ceiling.

Lineage is working with meat companies to store excess products in secondary sites in Oregon and Northern California, Mr. Beattie said. Workers in black-and-orange freezer suits work around the clock in zero-degree temperatures, packing away meat intended for China and other Asian markets.

NewCold, a cold-storage company operating warehouses in Washington and Idaho, has been fielding calls from meat companies, as well as vegetable and bread suppliers, all seeking available storage space, said business development head Jonas Swarttouw.

“There is so little slack in our cold storage industry that this little ripple immediately creates concerns,” Mr. Swarttouw said.

The supply buildup puts meatpackers in a bind. Each week, processing plants slaughter millions of chickens and hundreds of thousands of hogs and cattle. Deliveries for that meat are scheduled months in advance, with more livestock and poultry being raised to head to slaughter next. To keep ground beef and chicken breasts moving, meat companies need to continually turn over their inventory, which has a limited shelf life even when frozen.

“It’s the sell it or smell it principle,” said Christine McCracken, protein analyst for agricultural lender Rabobank. She said the epidemic prompted the cancellation in recent weeks of major springtime meat-buying meetings in China, South Korea and the Philippines, potentially further weighing on sales this year.

Mike Cockrell, Sanderson’s chief financial officer, said the company had planned for higher shipments of chicken-leg quarters to China this year. Sanderson could instead sell more of those products to Mexican buyers, reducing the need to store them, he said.

Meat industry officials hope the epidemic represents a temporary delay to the long-anticipated surge in exports to China. They will have more meat to market this year: the USDA projects red meat production will rise 2.9% this year, while poultry production is estimated to grow 3.9%. The U.S. dollar is another challenge. The virus’s spread has driven investors toward lower-risk assets, lifting the U.S. dollar, making U.S.-produced meat more expensive relative to competing products from other countries.

Meat producers can’t count on U.S. consumers to eat up the slack, said Altin Kalo, an economist with Steiner Consulting Group.

“We simply cannot absorb domestically all the protein,” he said. “If we really slow things down and we’re not able to export product, it’s going to be a dire situation.”