WASHINGTON — Speaker Nancy Pelosi said on Friday that the House would take action next week to address the baby formula shortage that has left parents desperately searching for food for their children, as President Biden pledged action that would result in more formula on store shelves within “weeks or less.”

In a letter to lawmakers, Ms. Pelosi said she would expedite a bill to grant emergency authority to the federal food assistance program for women and children to relax restrictions on the types of formula that can be purchased. About half the infant formula sold nationwide is purchased through benefits provided by the Special Supplemental Nutrition Program for Women, Infants and Children, or WIC. Loosening the rules could help ensure that recipients are able to buy whatever type is available.

“The babies are crying, and the babies are hungry,” Ms. Pelosi wrote. “So we must take urgent action to protect their health and well-being.”

She said her fellow House Democrats were also working on an emergency spending bill to “immediately address the infant formula shortage.” It was not yet clear how large the measure would be or where the funding would go, but aides said one proposal being considered was purchasing formula from other countries that have excess supply.

“We have to move with caution as well as speed,” Mr. Biden said at the White House on Friday, when asked whether the administration had responded quickly enough to a shortage that began in February. He called it the most urgent issue he was facing.

In terms of increasing imports, the president said, “we have to make sure what we’re getting is in fact first-rate product.”

Mr. Biden said the Food and Drug Administration was taking steps that would yield results in “weeks or less, getting significantly more formula on shelves.”

His quick timetable and Ms. Pelosi’s plans reflected a growing urgency to address the shortage, which has become a national crisis and a political challenge as Republicans work to weaponize the issue before the midterm elections.

The White House on Thursday announced a series of modest moves to help increase the supply of formula, including plans to increase imports and speed manufacturing.

Republicans have spent the past few days hammering Mr. Biden for the shortage, pointing to it as the latest example of Democrats being slow to address the most basic needs of American families, a central piece of their campaign message.

Republicans have latched onto a xenophobic talking point, amplified by Fox News and other conservative outlets, that Mr. Biden has prioritized undocumented immigrants over Americans by providing pallets of baby formula to detention centers on the southwestern border.

“American mothers and their babies shouldn’t suffer because of the #BidenBorderCrisis,” Representative Elise Stefanik, Republican of New York, posted on Twitter on Friday.

A White House official noted that it has been compulsory under the law since 1997 for border personnel to provide food, including baby formula, for detainees in their custody.

Democrats made #EliseStarvefanik a trending topic on Twitter on Friday, heaping criticism on Ms. Stefanik and other Republicans who have questioned the practice and noting that the alternative would be for the government to starve children in its care.

Jen Psaki, the White House press secretary, said that Mr. Biden was considering invoking the Defense Production Act to increase production.

Representative Carolyn B. Maloney, Democrat of New York, and Representative Raja Krishnamoorthi, Democrat of Illinois, who chair the House committee on oversight and the subcommittee on consumer policy, sent letters on Friday to the top four formula manufacturers requesting information about what they were doing to address the shortage.

They said they were also seeking documents from Abbott Nutrition relating to the conditions of its shuttered formula manufacturing plant in Sturgis, Mich., that led to recalls of several of its products after four infants were sickened; two of them died.

On Friday, Abbott said it would extend rebates for alternative products through the end of August, in response to a letter from Tom Vilsack, the agriculture secretary, requesting that they do so.

Emily Cochrane and Zolan Kanno-Youngs contributed reporting.