Apple Inc. is facing new questions from government officials in the U.S. and France about its handling of battery-related performance issues on iPhones, a sign that controversy over the problem continues despite the technology giant’s apology last month.

On Tuesday, Sen. John Thune (R., S.D.), chairman of the Senate Commerce Committee, pressed Apple for answers to a series of questions about how the company decided to throttle back iPhone processing performance in phones with older batteries.

In a letter to Chief Executive Tim Cook, a copy of which was viewed by The Wall Street Journal, Mr. Thune asked how Apple has tracked customer complaints of processing performance and if Apple has explored offering rebates to customers who paid full price for a battery replacement before the company offered discounted rates last month.

In France, the Paris prosecutor’s office said it has opened an investigation into Apple for potential deception and “programmed obsolescence.” The investigation—which could lead to preliminary charges or be dropped—will be run by the consumer protection agency that falls under the country’s finance ministry, a spokesman for the office said Tuesday.

Apple has been under fire since mid-December from customers and analysts saying they had noticed a slowdown in the performance of older iPhones.