US authorities upgraded an investigation into a possible defect involving Tesla Inc.’s top-selling models after identifying almost 2,400 complaints about drivers losing steering control.

The National Highway Traffic Safety Administration escalated its preliminary evaluation of the issue — started in July of last year — to an engineering analysis, according to a notice posted on its website Friday. The upgrade means the agency’s Office of Defects Investigation determined more testing was warranted, and is a step toward a possible recall.

The probe pertains to an estimated 334,569 Model 3 sedans and Model Y sport utility vehicles. In addition to disclosing 2,388 total complaints, NHTSA said it’s aware of more than 50 vehicles that had to be towed as a result of steering issues, and one crash.

Tesla supplied certain steering rack part numbers to investigators, suggesting the probe could lead to a physical recall, as opposed to an over-the-air software remedy. That would be significant because repairs would be costlier for the company.

Tesla didn’t immediately respond to a request for comment on the notice.

The probe is one of several that are pending for the carmaker led by billionaire Elon Musk. NHTSA also has been looking into Tesla seat-belt failures, and has kept open an investigation into the company’s driver-assistance system Autopilot to monitor the efficacy of a fix deployed late last year.