Originally introduced in 2017, the bill passed the House unanimously in 2020 only to stall in the Senate. It faced headwinds earlier this year after progressive Democrats in Congress, led by Representative Cori Bush of Missouri, said they would oppose the bill, citing concerns from groups including the American Civil Liberties Union and the Council on American-Islamic Relations that it would increase surveillance of people of color, especially activists.

The A.C.L.U. opposed an earlier version of the legislation, stating in a 2019 letter that it would “entrench longstanding problems, and result in the further unjustified and discriminatory surveillance, investigation, and prosecution of people of color and other marginalized communities, including those engaged in First Amendment protected activities.”

But the measure has picked up momentum since Saturday, when an 18-year-old white gunman opened fire at the Tops supermarket in East Buffalo, authorities said, in a premeditated effort to kill Black people, driven by the belief that white Americans were at risk of being replaced by people of color.

Democrats negotiated internally to amend the bill to assuage the concerns of progressives, narrowing the definition of domestic terrorism and adding a provision to guarantee that individuals could not be put under surveillance for the mere act of taking part in a protest.

Attorney General Merrick B. Garland and Homeland Security Secretary Alejandro N. Mayorkas testified before Congress last year that the greatest domestic threat facing the United States came from what they called “racially or ethnically motivated violent extremists.”

In a separate warning last year, the Department of Homeland Security said publicly for the first time that the United States faced a growing threat from “violent domestic extremists” emboldened by the Jan. 6 attack on the Capitol.

Senate sponsors of legislation framed the bill last year as a way to respond to the assault on Congress.

Several members of Congress, particularly Democrats facing difficult re-election races, have clamored in recent days for Mr. Biden to invoke the law to ease the formula shortage. It was not clear how quickly the impact would be felt.

A White House official, speaking on the condition of anonymity and without authorization to comment on the plan, said it could take time for companies to make use of the Defense Department’s offer to ship formula. But the official added that Mr. Biden was determined to do anything he could to shave even a few days off the manufacture and distribution of formula.

The White House announced the effort as it confronted aggressive questioning by Congress. In recent days, lawmakers have announced plans to haul administration and industry officials to Capitol Hill for testimony, demanded answers from Mr. Biden’s team on how the shortage was allowed to develop, and launched investigations into the crisis and Abbott Nutrition, the company that recalled several of its formula products after at least two infants died.

On Wednesday evening, the House overwhelmingly approved bipartisan legislation that would grant the federal government emergency authority to expand the types of formula that can be purchased with benefits from the Special Supplemental Nutrition Program for Women, Infants and Children, known as WIC. About half the formula sold in the United States is purchased with WIC benefits, and the legislation aims to ensure that those benefits can be used to buy whatever formula is available.

The measure was expected to move quickly through the Senate. But the proposal to issue emergency money for the F.D.A., which passed the House 231 to 192 on Wednesday night, faced a more difficult road amid opposition from Republicans, who said it was a blank check for an agency that had failed to prevent a major crisis.

In a notice sent to rank-and-file lawmakers, Republican leaders charged that Democrats had abandoned bipartisan talks and instead put forward legislation “with no plan to actually fix the problem, all while failing to hold the F.D.A. accountable.”

The WIC measure, by contrast, drew broad support, passing 414 to 9.

“I know what it means to not be able to stretch out milk for the entire month,” Representative Jahana Hayes, Democrat of Connecticut and a sponsor of the bill, said this week. Ms. Hayes, who said she had used WIC benefits to purchase formula, added, “with everything going on right now, the last thing a family should have to worry about is feeding their children.”

Mr. Adams introduced a new color-coded alert system in March to let New Yorkers know when they are at greater risk. On Tuesday, the city entered the orange, or “high” risk level, which comes with the recommendation that the city require face masks in all public indoor settings. The guidelines also recommend that Mr. Adams consider requiring masks at schools and bring back a proof-of-vaccination requirement at restaurants and gyms.

Mr. Adams said that the situation had changed since March, and that he was focused on distributing antiviral medications and home tests.

“We’re staying prepared and not panicking,” Mr. Adams said. “When I look at the hospitalizations and deaths, the numbers are stable.”

Earlier this week, New York City health officials strongly recommended that all individuals wear medical-grade masks in offices, grocery stores, schools and other public indoor settings. Mr. Adams has echoed that advice, but he said earlier this week that he would not consider mandates until “there comes a time that our hospitals are in a state of emergency, or we’re trending that way, and my doctors that run the hospitals tell me this is what we need to do.”

Mr. Adams has received criticism for not moving aggressively enough to contain the virus, including from Dr. Jay Varma, a senior health adviser to former Mayor Bill de Blasio. Asked on Wednesday about that criticism, Mr. Adams thanked Dr. Varma for his service, but said that he would set his own policy.