Tesla Inc., with a fresh civil fraud settlement now behind it, faces a new legal problem: a deepening criminal investigation.

Federal Bureau of Investigation agents are examining whether Tesla misstated information about production of its Model 3 sedans and misled investors about the company’s business going back to early 2017, people familiar with the matter say.

Action in the criminal investigation, headed by the U.S. attorney’s office in San Francisco, has intensified in recent weeks after the Securities and Exchange Commission settled separate civil charges with Tesla and Chief Executive Officer Elon Musk, the people said.

Tesla had disclosed on Sept. 18 that it had received a “voluntary request” for documents from the Justice Department, 10 days before the company and Mr. Musk struck a settlement with the SEC of civil charges on in a separate case involving controversial tweets from Mr. Musk. But it hasn’t been previously reported that the Justice Department is focusing on Tesla’s Model 3 production issues dating to early last year and that the criminal securities-fraud probe is intensifying.

Tesla didn’t immediately respond to a request for comment. The Justice Department and SEC declined to comment.

On Wednesday, Tesla reported record quarterly profit as increased output of its Model 3s ended seven quarters of losses for the electric-car maker. The third-quarter results, which buoyed Tesla’s stock, followed a series of production delays. Mr. Musk is betting that the Model 3 can transform the company from a niche luxury brand into a legitimate rival against the world’s largest auto makers.

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In recent weeks, FBI agents have contacted former Tesla employees asking them for testimony in the criminal case. The former employees received subpoenas earlier in the probe, and FBI agents recently have sought to interview a number of them, the people said.

In February 2017, after reporting fourth-quarter 2016 results, Tesla laid out an aggressive production plan to bring out the Model 3, with plans to ramp up to 5,000 vehicles a week in the fourth quarter. On a conference call that month with analysts, Mr. Musk said he was pushing suppliers to be ready for a weekly run rate of 1,000 vehicles in July to 2,000 in August and 4,000 in September.

A few months later in July, Mr. Musk sounded confident that Tesla would be producing 20,000 Model 3s a month in December 2017, in line with his previous pledge of having 5,000 vehicles a week by year’s end. “Looks like we can reach 20,000 Model 3 cars per month in Dec,” he tweeted on July 2, 2017, days before the first Model 3 rolled off the production line.

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In the early weeks of production, the company’s body shop, where the skeleton of the cars would take shape, wasn’t fully functional, according to an October 2017 article in The Wall Street Journal. Tesla was still hand-building parts of the Model 3s, and the body shop wasn’t fully installed until September, according to people quoted in the article that were familiar with the situation.

Tesla ended up producing 2,700 Model 3s for all of 2017, and 793 in the last week of 2017.

Now the FBI is comparing the company’s statements with its production capability during 2017. Authorities are homing in on whether the company made projections about its Model 3 production knowing it would be impossible to meet the goals, people close to the situation say.

Model 3s began assembly in July 2017 and Tesla finally reached the long-promised goal of making 5,000 of the vehicles in a single week during the last seven days of June 2018. The company aimed to build on that pace but ultimately averaged 4,300 Model 3s a week, excluding planned downtime, during this year’s third quarter. It said it built more than 5,300 of the cars during the final week of the period.

Though Tesla shares have surged this week on its profit picture, they still are down roughly 17% since a high in June 2017.

The SEC isn’t involved in the investigation into production issues at Tesla, people familiar with the matter said.

The SEC settlement related to a statement by Mr. Musk through his Twitter account on Aug. 7 that he had “funding secured” to take Tesla private. The SEC alleged that the tweet by Mr. Musk was “false and misleading.”

Under the settlement, Mr. Musk and the company agreed to pay a total of $40 million in fines and to remove Mr. Musk from his role as chairman of the board, without admitting or denying wrongdoing.

The Justice Department and SEC often coordinate investigative efforts, but the SEC’s recent civil action over Mr. Musk’s tweet on taking the company private didn’t address questions about Model 3 production disclosures.