Doughnut shops, once a rarity here, have proliferated across the city, with a huge number of rivals — including American giants Dunkin’ Donuts and Krispy Kreme — now battling for supremacy in the race to give Shanghai’s middle-class consumers their morning coffee-and-sugar fix.

The opening of so many doughnut shops in so many locations is a testament to Shanghai’s growing affluence and the belief that young Chinese with more disposable income will be hankering for more leisure food. As China has opened its doors to the world, American fast-food chains such as KFC, McDonald’s and Starbucks have exploded here, mostly in wealthier cities and coastal areas, with varying degrees of success.

So an American-style doughnut, with a cup of coffee, would seem a perfect fit for China’s modern, on-the-go city lifestyle.

But others look at the swelling number of doughnut shops here wonder if it’s just too much. Some shops have disappeared. Many others appear mostly empty at peak hours — and Chinese customers seem more interested in the drinks than the sugary doughnuts. And following the lessons of other American retailers, the doughnut shops are finding that some of their best-sellers would be barely recognizable back home, like Dunkin’s dried pork and seaweed doughnut, or the doughnut made with dried Bonito fish.

Retail industry analysts think Shanghai’s once-languid doughnut market might already be saturated. Some speak of an ensuing “doughnut war,” which might leave just a few survivors.

“We’re going to need a U.N. resolution very soon — they’re going to have to declare a sugar-free zone over Shanghai,” said Paul French, the British-born founder of a market research company, Access Asia, that focuses on the retail sector. “There’s too many, because we’re starting to see them close down.”

Dunkin’ has the most ambitious plans. The company opened its first store in China in November 2008, and announced plans to have 100 across the country within 10 years. There are now 40 in China, with 18 here in Shanghai, according to Frederick Sze, Dunkin’s managing director for greater China.

“We are number one in the U.S., and we want to be number one in China also,” Sze said.

Krispy Kreme, Dunkin’s main U.S. rival, came one year later, and snagged a prime location on a popular pedestrian restaurant street just off fashionable Nanjing Road. The 3,000-square-foot, two-story store has couches, wireless Internet and an open viewing window onto its “doughnut theater,” showing the sugary dough rings rolling off a conveyor belt.

Local chief executive Jujing Lim said there are plans for three new Krispy Kreme outlets in Shanghai, and his team is still scouting for the best locations.

The oldest of the group is Mister Donut, a brand which largely disappeared from the United States but remains strong in Asia under its Japanese corporate owner. Mister Donut has been in Shanghai since 2000 and has eight outlets in the city. And there’s an Australian usurper, Donut King, which arrived in 2008 and now has 11 stores in Shanghai, plus several local versions of the doughnut shop, such as the Taiwanese-run Cafe 85 C.

But what isn’t at all clear is whether Chinese consumers particularly like doughnuts.

The average Chinese breakfast might consist of congee, or rice porridge, maybe some soybean milk, sometimes fried noodle, or perhaps a dry roll or bun. The idea of something as sweet as a glazed or cream-filled doughnut in the morning would seem an anathema to many local palates.

“I’m not a big doughnut lover, and I only have one once a month,” said a 28-year-old woman working as a marketing manager, who stopped by Krispy Kreme on a recent Friday. “There are too many calories, and they’re too sweet, unacceptably sweet, especially the chocolate ones. But the doughnut looks really cute!”

Several of the doughnut shops appear empty in the mornings, when they should see heavy traffic.

Dunkin’, like some of the other chains, is discovering that coffee and other drink offerings, including jasmine green tea and lichi green tea, are more popular than doughnuts.

Krispy Kreme, meanwhile, is offering its quarters, with easy chairs and quiet surroundings, as a place to relax, surf the Web and enjoy a huge variety of cream-filled doughnuts at a more leisurely pace.

“People stay a long time,” Lim said. Here in Shanghai, he said, “we position ourselves differently than in the West.”

Still, the pessimists think the doughnut might have a hard time finding a toehold in China — as evidenced by the largely empty doughnut stores, and the number of leftovers on the shelves at closing time. “It’s one of those food concepts that has singularly failed to set the country alight,” said French, the retail analyst.

French noted the biggest obstacle yet: In Shanghai, police officers seem to prefer smoking cigarettes to taking a doughnut and coffee break. “They haven’t cracked the cop market,” he said.

Researcher Wang Juan contributed to this report.