WASHINGTON—The Trump administration backed a plan to send checks directly to Americans as part of a $1 trillion stimulus package to help households and businesses, a dramatic step designed to cushion the impact of the sudden economic slowdown triggered by the coronavirus pandemic.

Treasury Secretary Steven Mnuchin pitched Senate Republicans on a stimulus package that would include an initial $250 billion for direct payments, according to administration officials and lawmakers, part of a wide-ranging fiscal and monetary effort. One of the officials said the administration would push for additional direct payments beyond the $250 billion in the coming weeks, if needed.

“It is a big number. We’ve put a proposal on the table that would inject a trillion dollars into the economy,” Mr. Mnuchin told reporters at the Capitol.

The stock market rallied Tuesday, with the Dow Jones Industrial Average rising 5.2%, rebounding from a 13% drop a day earlier. The steps outlined by the Trump administration followed similar efforts in Europe, where France and other countries pledged tens of billions of euros in immediate aid for businesses and employees.

All of government stepped up its response. The Federal Reserve said it would start making loans to American corporations, relaunching a financial-crisis-era commercial-paper tool to help calm short-term debt markets. The Pentagon said it would provide up to five million respirator masks to safeguard front-line responders, while the Treasury Department said it wouldn’t penalize late tax payments by many Americans.