BEIJING— Chance Jolivette doesn’t mind that the sneakers he orders online are fake.

The 33-year-old aspiring rapper in Gueydan, La., is a connoisseur of counterfeits, and among a rising number of people who post videos critiquing their quality in a twist on the “unboxing” videos shot by fans of new products. His source of choice: AliExpress, the global sales platform of Chinese e-commerce company Alibaba Group Holding Ltd.

“Shoe collecting has become a culture within itself and it is very expensive, so most people aren’t able to indulge,” said Mr. Jolivette. “That is why I find the cheaper shoes on AliExpress and try to show people the pros and cons.” Since spring he has spent more than $1,000 on 20 pairs of fakes, including knockoffs of Nike’s Michael Jordan and LeBron James lines.

Nike Inc. declined to comment.

Alibaba, which dominates e-commerce in China, is going abroad as its home market matures. Alibaba Executive Chairman Jack Ma has said the company plans to serve two billion customers world-wide within a decade. Executives hope to turn China’s Nov. 11 online sales festival, Singles’ Day, into a global event.

But the presence of counterfeit goods presents a problem for these global ambitions. Business groups and brands are increasingly scrutinizing AliExpress, worried it will send counterfeits from Chinese factories straight into consumers’ hands on a global scale.

“There are substantial concerns among brand owners that expanding Alibaba’s AliExpress platform will result in a very significant increase in the sale of counterfeit and otherwise infringing goods” unless more safeguards are imposed, said Paul Kilmer, founder of the Trademark Working Group, a collective of U.S. companies and organizations.

Alibaba says counterfeiting is a problem for all e-commerce companies, and that only a minuscule proportion of transactions lead to consumer complaints about fakes. “Like all global companies in our industry, we must continue to do everything we can to stop these activities,” a spokesman said. He added Alibaba has made it easier for brands and others with a record of legitimate requests for the removal of infringing listings to use a simplified system to expedite such take-down claims.

AliExpress connects global consumers with Chinese wholesalers and manufacturers, primarily small and midsize businesses, through sites in English, Russian, Spanish, Bahasa Indonesia and other languages. Sellers pay Alibaba commissions on transactions as well as for advertising on the marketplace. In the fiscal year ended March 31, AliExpress generated $6.6 billion in gross merchandise value, a measure of total transactions on the platform.

The platform has led Alibaba’s international growth. Revenue from overseas retail in the fiscal year ended last March was up 88% from the year ended March 2014, to $285 million, which in turn was nearly 2½ times that of the year before. AliExpress is rapidly adding shoppers, particularly in Russia, Brazil and the U.S., according to the company.

But in a lawsuit against Alibaba, filed in May, French luxury group Kering SA said AliExpress “knowingly provides a marketplace for merchants engaged in the sale of the counterfeit products.” Alibaba has said the complaint has no legal basis and that it will fight the lawsuit.

In a September filing with the U.S. Trade Representative over China’s compliance with World Trade Organization commitments, the U.S. Chamber of Commerce cited AliExpress as a source of concern.

“AliExpress is expanding into the U.S. and offering translation services in multiple languages,” the chamber said in its submission, and neither the rights holders nor the government has the resources to stop a flood of counterfeits coming in one small package at a time.

Alibaba said it has built relationships with government and industry bodies to combat counterfeiting and continues to improve its own monitoring system.

Alibaba’s investor-relations chief, Jane Penner, said at a conference in September that sellers on AliExpress are increasingly competing on price, which could reduce quality. “We’re working now to upgrade AliExpress to a platform with both higher-quality merchants and higher-quality products,” she said.

Counterfeit merchandise could also hurt AliExpress’s image with consumers. In southern Slovakia, 56-year-old Lubomir Sajbidor said he has successfully bought car accessories, a baseball cap, wheels for a shower door and tomato seeds. But a Huawei Technologies Co. smartphone he bought for $132 turned out to be fake, as he discovered when a case he bought didn’t fit.

Alibaba couldn’t help him track down the vendor who sold it to him, he said. “I was satisfied with them, but with this phone, they made me mad,” Mr. Sajbidor said. Huawei declined to comment. Alibaba said it is looking into the matter.

Jeremy Pyles, co-founder of Niche Modern, an upstate New York-based manufacturer of handblown glass lights, says merchants on AliExpress and other sites have used his copyrighted photographs to sell knockoffs of his lights. Alibaba said it is looking into Niche Modern’s complaints.

Mr. Pyles said he now has an employee dedicated to filing complaints to such websites. “If AliExpress makes a huge foray into the U.S. market, let me tell you, I’m going to have people on that site day in, day out,” he said.

—Kathy Chu in Hong Kong contributed to this article.