That owner happens to be Chinese conglomerate HNA Group Co. , a controversial actor on the global scene, which in the spring built its stake in the German lender to nearly 10%.

Mr. Cryan has told associates he wanted nothing to do with the Chinese conglomerate. The iciness has raised eyebrows among Deutsche Bank supervisory-board members and clients, say people close to the bank. And it has irked Paul Achleitner, the company chairman who helped woo HNA. Mr. Achleitner doesn’t like executives airing negative views of individual shareholders, people close to the bank say.

The tensions between HNA, Mr. Cryan and Mr. Achleitner come as Deutsche Bank has struggled for most of a decade to revive profits and bring stability in its upper ranks. Undermining those rebuilding efforts are a slow turnaround and internal clashes over style and strategy that have spilled into meetings with clients and regulators, according to people close to the bank.

At the center is Deutsche Bank’s chief executive, Mr. Cryan, a Briton overseeing Germany’s largest lender. Mr. Cryan has had to mediate between German and U.K. managers vying for money and staffing in the bank’s reconfiguration for a post-Brexit world. He’s under pressure from investors to cut costs, update technology and resuscitate investment-bank profits. And in the background, he has shunned HNA.

Mr. Achleitner was the primary architect of Deutsche Bank’s existing executive suite and is known as a hands-on chairman. Some senior executives at times have bypassed Mr. Cryan to appeal to Mr. Achleitner for support on high-level decisions, people close to the bank say.

Early on, this muddied the management waters for Mr. Cryan, people close to the two men say. A bank spokesman says it is normal for a German chairman to have “regular interaction with all management board members.” Mr. Achleitner supports Mr. Cryan, and it is good for them to disagree at times, some of the people close to them say.

Over the past year, Deutsche Bank has survived legal battles that threatened its stability in the eyes of investors and clients, and raised $8.5 billion in capital. It also has flip-flopped on major strategy decisions, such as splitting and then recombining its trading unit and investment bank.

But this year brought other, fresh complications. Mr. Achleitner helped Deutsche Bank court HNA, before and during the German lender’s capital hike in March and April, people close to the bank say.

Mr. Cryan and HNA CEO Adam Tan haven’t met, the people close to the bank say. Mr. Achleitner and HNA representatives tried to bring the two together, but Mr. Cryan prefers to focus on clients and regulators, people close to the Deutsche Bank CEO say.

“Yes, he hasn’t met him, but of course he will,” a Deutsche Bank spokesman said.

HNA has drawn scrutiny well beyond Deutsche Bank. German and European regulators are reviewing the company, its funding and its Deutsche Bank holding, including what influence HNA might have over the bank as its largest shareholder. The closely held company had faced mounting questions about its ownership and potential ties to the Chinese government as it has ramped up overseas investment.

In the U.S. and elsewhere, some investment banks eager to profit from HNA’s deal fervor by funding and advising on more of its acquisitions have hit pause, people close to the firms say.

HNA uses derivatives to limit losses and gains on its Deutsche Bank holding. Mr. Cryan views the structure HNA used to invest in the lender as speculative, saying it creates risks for the lender and investors, people close to the bank say.

HNA said through a spokesman that it “continues to view Deutsche Bank as an attractive investment opportunity” and is “committed to supporting Deutsche Bank’s long-term success.” The company said it complies and cooperates with regulatory authorities and its relationship with banks remains strong. HNA has taken steps to restructure its ownership, including by setting up a New York foundation intended to be its biggest shareholder. The company spokesman said the foundation “looks forward to sharing more about its activities and ambitions in time.” A person familiar with HNA said the Chinese government doesn’t own a stake in HNA and none of its shareholders hold shares for any Chinese government officials.

HNA first disclosed in mid-February that it held a 3.04% stake in Deutsche Bank through Austrian asset manager C-Quadrat Investment AG . On March 5, Deutsche Bank announced plans to raise $8.5 billion through a share sale. HNA’s widely known interest in increasing its stake helped the capital hike go smoothly, by signaling support from a big new shareholder, bankers and investors say. This can encourage other potential new investors to follow suit.

On March 14, Mr. Tan had dinner in Frankfurt with Mr. Achleitner, Deutsche Bank’s then-Chief Financial Officer Marcus Schenck and a C-Quadrat executive, according to people familiar with the dinner. At the time, the HNA contingent expected Mr. Cryan to attend, some of the people say. He ended up not being in Frankfurt that night, according to people familiar with his schedule.

Two days later, Deutsche Bank announced that its supervisory board would nominate two new members, including one proposed by HNA: C-Quadrat’s CEO, Alexander Schütz. Mr. Schütz was advising HNA on other potential deals and had sold part of his firm to HNA.

In early May, HNA disclosed details of its nearly 10% Deutsche Bank holding in a regulatory filing. It showed that HNA used more than $2.8 billion in financing, mainly from UBS Group AG, to help purchase shares worth about $3.8 billion at the time. It also did a derivative deal with UBS. One portion of that deal protected HNA if the shares fell below €15. They have been below €15 since early August and closed Friday at €14.72.

Most Deutsche Bank executives learned about the structure from the filing, the people close to the bank say. HNA acquired most of its shares in the market, limiting Deutsche Bank’s control over the process, according to regulatory filings and the people close to the bank.

Mr. Cryan, a former UBS finance chief, told some people close to the bank that it couldn’t be certain how HNA is funded and that he didn’t like that the investor bought the bank’s shares and effectively bet against them.

On May 18, at Deutsche Bank’s annual meeting, shareholders overwhelmingly voted Mr. Schütz onto the supervisory board. While in Frankfurt, the board gathered for dinner. Afterward, chauffeured cars waited to take them to their homes and hotels.

Mr. Schütz was perplexed when no car appeared for him, and he caught a ride with a fellow board member, according to a story he and others recounted over the next few days. The ensuing banter was that someone at Deutsche Bank must have wanted the HNA guy to walk. Another version of events is that Mr. Schütz had declined a car, so Deutsche Bank planners hadn’t booked one.

Mr. Schütz tried again to arrange a dinner between Messrs. Cryan and Tan, suggesting they could meet in New York. Mr. Cryan didn’t respond to the invitations, people familiar with the matter said.