Securities regulators began investigating last year whether Tesla Inc. misled investors about its Model 3 car production problems, according to people familiar with the matter.

The Securities and Exchange Commission subpoenaed a parts supplier for the auto maker as part of the probe, one of the people said, well before the regulators began looking into Elon Musk’s tweet last week about taking the company private.

Regulators have also subpoenaed Tesla’s directors seeking to learn what they knew about Mr. Musk’s plan to take Tesla private, according to another person familiar with the matter.

Both issues are being handled by the SEC’s San Francisco office, one of the people said.

The Wall Street Journal reported in October that the Tesla assembly plant’s body shop wasn’t fully installed until around September and that major portions of the Model 3 were being hand built weeks after Mr. Musk announced production had begun in July 2017. As production started, he claimed about 1,600 cars would be made in the third quarter of 2017 before reaching 20,000 in December. Those forecasts were far below what he predicted roughly a year earlier, when he said as many 200,000 Model 3s would be made in the second half of 2017.