WASHINGTON—The government is weighing a plan to rescue Citigroup Inc., whose stock has been hammered on worries about its financial health. The Treasury Department and the Federal Reserve have been in discussions over the weekend to devise a strategy to stabilize the company, according to people familiar with the talks. They spoke on condition of anonymity because the discussions were ongoing.

What investors are worried about is that all the risky debt sitting on Citigroup’s balance sheet will eventually turn into losses as the economy worsens and the markets stay turbulent—losses that could be nearly impossible to reverse.