As Volkswagen AG faces a Friday deadline for proposing how it will fix nearly half a million cars tainted by illegal software, the German auto giant is grappling with a contentious relationship with regulators in the U.S. and Europe who will determine the penalties for its emissions-cheating.

Two months after the U.S. Environmental Protection Agency disclosed that Volkswagen deceived diesel emissions testers for years and later falsely claimed technical problems when challenged, the details of how and why Volkswagen cheated, who was responsible and how it plans to fix affected cars are still largely unclear.

Volkswagen is set to discuss initial recall plans in meetings Thursday and Friday with U.S. and California regulators, an EPA spokeswoman said. On Friday, the company also is expected to provide details of deep spending cuts.

Since the EPA disclosure, Europe's largest car maker has publicly sparred with U.S. regulators over details, angered German and European Union officials by failing to provide information and frustrated investors. Its stock-market value has declined 37%, and the company has set aside almost $10 billion to pay for the crisis.

Buyers also have cooled on Volkswagen cars. In October, the first full month of sales since the crisis became public, Volkswagen brand sales fell more than 5% world-wide, while the U.S. and European car markets grew.

"I'm disappointed…by their response so far," said Mary Nichols, head of the California Air Resources Board, one of the agencies meeting with Volkswagen this week. "I would have hoped we would have heard from them sooner and in a more forthright fashion," she added.

Volkswagen officials say their initial response to the crisis was slowed by management upheaval and the scope of the scandal, but has since improved. "There is no way to have a plan for a crisis like this," said spokesman Hans-Gerd Bode. "Nevertheless, we responded quickly, and very quickly said how we would get to the bottom of what happened."

A more than $1,000 cash-and-services offer to U.S. diesel-car customers this month that was meant to show goodwill instead met a lukewarm response and scorn from some U.S. lawmakers, who called it "insultingly inadequate."

"This is a textbook example of how it should not be done," said Sasja Beslik, head of Responsible Investments at Swedish fund group Nordea Asset Management. "Volkswagen's communications have been totally reactive." Nordea held about half a million shares of Volkswagen's common stock before the crisis, but since has sold off 90% of its holdings, Mr. Beslik said.

Impressions matter in high-stakes government investigations. Past cases indicate the U.S. is more inclined to impose lower fines on companies that are seen as cooperative in public, even if disagreements over facts continue behind closed doors.

In its settlement with General Motors Co. over a defective ignition switch linked to more than 100 deaths, the U.S. Justice Department credited the auto maker for a swift and robust internal probe that furnished federal prosecutors with documents ordinarily protected by attorney-client privilege. Prosecutors also noted GM's decision to fire wrongdoers and set up a compensation fund to dole out hundreds of millions of dollars to victims.

GM paid a $900 million financial penalty from prosectors, less than the $1.2 billion assessed Toyota Motor Corp. in a criminal settlement for misleading consumers and concealing safety problems in its vehicles.

Prosecutors had difficulties interviewing Toyota executives in Japan because the company refused to make them available, people familiar with the matter have said. Toyota has said it is working hard to improve communications with customers and regulators. Toyota also cooperated with the probe, prosectors said in settlement documents.

Volkswagen's scandal, potentially affecting more than 12 million vehicles world-wide, would swamp most businesses.

No one holds more sway over how much Volkswagen eventually may have to pay in fines and penalties than the EPA. But Volkswagen often has been defiant, to the point of publicly criticizing the agency's testing methodology and suggesting the EPA was out to stifle competition.

Days before offering the U.S. goodwill package, Volkswagen received another notice from the EPA, this time alleging cheating software also was in 3.0-liter diesel engines used in luxury sedans and sport-utility vehicles made by its namesake, Audi and Porsche units. Volkswagen contested the EPA's claim, but still angered some customers who bought the vehicles only to see them drop in value.

"They should have pulled it off the market months earlier," said Charles Taylor, a 74-year-old retired college athletic director in Brevard, N.C., after he found out that a 2015 Audi Q7 sport-utility vehicle he had just leased was among the tainted vehicles. Audi halted sales of the model with a three-liter diesel engine in the U.S. even though it wasn't among the specific ones the EPA cited.

Mr. Taylor later returned the vehicle and received a full refund from the dealership, he said. He recently leased a new Acura MDX SUV, he said.

Some dealers say they are equally frustrated.

"We still don't know what to tell our customers," said Steve Kalafer, who co-owns Flemington Car & Truck Country, a New Jersey-based dealership. "We need details about how VW plans to make our customers whole."

Interactions between Volkswagen and the EPA got so bad that German Transport Minister Alexander Dobrindt publicly criticized the company after he met with agency officials last month.

The EPA "is dissatisfied—angry," he said in Washington, D.C., as he left the meeting. "Trust has been destroyed and it will take considerable work to rebuild it."

An EPA spokeswoman declined to comment on Wednesday.

CARB's Ms. Nichols said she hopes her staff this week will hear clear plans from Volkswagen for fixing the vehicles and what incentives or other measures it would offer consumers to prod them into getting repairs completed. Regulators will review Volkswagen's plans and continue discussions with the auto maker, she said.

Volkswagen's U.S. chief executive, Michael Horn, said his company is cooperating fully with U.S. and California regulators and aiming to be as transparent as possible. "We understand apologies are not enough," Mr. Horn said on Wednesday in Los Angeles.

"Nothing is more important to me personally than…making things right," he said. "There is frustration and sometimes anger. All these reactions are fully understandable."

Mr. Horn said 120,000 customers have so far registered for the goodwill cash-and-services offer Volkswagen recently launched.

Volkswagen has tapped law firm Jones Day to conduct an investigation and invited German prosecutors to scrutinize its activities. It also hired an international team of public-relations firms and recruited an "Integrity Officer" from rival Daimler AG to help the company rewire its corporate governance and corporate culture.

Under new Chief Executive Matthias Mü ller, who took over from the ousted Martin Winterkorn on Sept. 25, Volkswagen is offering immunity from job discipline to employees who come forward this month with additional information about the emissions deception.

This week, Volkswagen showed a German parliamentary committee of experts how it planned to fix affected 1.6-liter vehicles. The fix, which the company hasn't explained to the public, is now undergoing further tests by Germany's KBA, the motor vehicle agency, an agency spokesman said, declining to provide any details.

Volkswagen also has frustrated regulators in Europe.

EU Industry Commissioner Elzbieta Bienkowska was in Berlin recently and believed she had a meeting set with Herbert Diess, chief of the Volkswagen brand. The meeting was canceled.

"I don't think it was a good decision," she said. "I don't know why it was canceled. It was not me who canceled."

A Volkswagen spokesman said Mr. Diess and Ms. Bienkowska didn't have a confirmed appointment that day. While possibly a misunderstanding, the exchange shows how tense Volkswagen's relationship with Brussels has become.

Even Volkswagen's closest allies in Germany have complained the company left them uninformed as the crisis brewed.

Lower Saxony Prime Minister Stephan Weil told his legislature on Oct. 13 that he and a top deputy heard the news "from media reports" over the weekend. Volkswagen's handling was "absolutely unacceptable," he said later. "In my opinion, the information should have been available a lot sooner."

The German state holds 20% of Volkswagen's voting stock and depends on Volkswagen and the automotive industry for 200,000 jobs. Mr. Weil sits on the company's supervisory board.

A Volkswagen spokesman said management wasn't required to inform Mr. Weil or other directors until the full dimension of the scandal became clear.

German Vice Chancellor Sigmar Gabriel, a former director on Volkswagen's board, also learned about the scandal from media reports. When one of his top aides awoke to coverage on Saturday morning, Volkswagen was unreachable, the aide said.

Volkswagen made a brief statement on Sunday, but as Mr. Gabriel prepared to give a speech that Monday at the Frankfurt Motor Show, he still had few details about the storm threatening Germany's largest industry.

"We came in Monday and the house was on fire," the aide said.

Harriet Torry, Friedrich Geiger and Ruth Bender contributed to this article.

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November 18, 2015 19:55 ET (00:55 GMT)