WASHINGTON — Republicans on Capitol Hill offered a preview this week of how they plan to attack the Biden administration’s immigration policies as the midterm elections approach, trying to make the homeland security secretary, Alejandro N. Mayorkas, accept blame for a historic spike in migration across the southwest border.

During more than eight hours of testimony and at times heated exchanges over two days of hearings on Capitol Hill, Mr. Mayorkas did no such thing.

“I feel that America needs you to own this thing, good sir,” Representative Clay Higgins, Republican of Louisiana, said on Wednesday during a hearing conducted by the House Homeland Security Committee. “Speak to the American people and own this thing. We’re losing our country down there. We need you to resign.”

During a House Judiciary Committee hearing on Thursday, Representative Ken Buck, Republican of Colorado, said his constituents think Mr. Mayorkas has committed treason and should be impeached.

Mr. Mayorkas shot back, “What you have just said is so profoundly offensive on so many different levels, in so many different regards. I won’t ask you for an apology.”

The stark ideological differences over immigration are nothing new, but a once-obscure public health rule used to restrict immigration during the pandemic has become the face of the issue as its end date, May 23, approaches.

Although the stated purpose of the rule, known as Title 42, is to limit the spread of the coronavirus in Border Patrol facilities and border towns, Republicans and even some moderate Democrats now see it as a potent tool for controlling immigration as Congress continues to punt on passing comprehensive changes to the country’s immigration laws.

Several Republican-led states have filed a lawsuit seeking to keep the public health rule in place.

The number of migrants who crossed the southern border without documentation has increased since President Biden took office. March set a record for the most crossings in a single month in decades: 221,303. The inflow has at times overwhelmed border officials, who lament being stuck indoors doing the paperwork to process migrants instead of patrolling the hundreds of miles of border the service is responsible for protecting.

Despite the public health rule, the Biden administration has allowed hundreds of thousands of migrants into the country to face deportation proceedings, mostly families with young children. The Trump administration also allowed in large numbers of migrants during a spike in migration in 2019. Many new arrivals apply for asylum, a process that can take six to eight years but that allows them to wait in the United States for a decision.

Title 42 gives Border Patrol agents the authority to expel people without asking whether they are afraid to return to the country they came from, a process that takes significantly longer.

Lifting the rule, critics say, will lead to a collapse of operational control along the southwest border.

When it was put in place at the beginning of the pandemic, the end was presumed to be in sight for the public health emergency, which would trigger the lifting of the rule. It has been long assumed that lifting it would drive up the number of undocumented migrants crossing the southwest border.

Earlier this year, the Biden administration was under intense pressure from Democrats to lift the rule, since the pandemic had reached a phase with relatively few hospitalizations and cases of serious illness.

At the end of March, the Department of Homeland Security released a 16-page operations plan for the southwest border that laid out how the government would respond to an increase in migration; the plan said officials were preparing for as many as 18,000 migrants a day.

Days later, the Centers for Disease Control and Prevention announced it would lift the emergency rule later this spring, drawing outrage from Republicans as well as some Democrats whose seats are vulnerable in the midterm elections.

In the weeks that followed, lawmakers from both parties demanded to see a plan for how the government would respond to a spike in migration without being able to quickly expel people under the public health rule; it was not clear whether they had not seen the Homeland Security Department’s plan or were merely dismissive of it. But their demands led to high-level hand-wringing in the White House over whether lifting the order on May 23 could lead to losing control of the House and Senate next year.

A new 20-page memo further outlining the plan was drafted and released this week, ahead of Mr. Mayorkas’s appearances on Capitol Hill, to discuss the department’s budget request for the 2023 spending year. The memo included a strategy broken down into six “pillars,” as well as 17 pages of background about the dysfunctional immigration system that the Biden administration inherited and a summary of steps the administration has taken to fix it.

The pillars include mobilizing more personnel and support at the border; taking new measures to speed processing of migrants to prevent overcrowding at border stations; moving to swiftly remove migrants who are not allowed into the country, as was done before the pandemic; working with nonprofit organizations that can help migrants who are allowed to stay in the country after they have been released from government custody; targeting smugglers; and working with other countries in the region to try to limit migration.

The Senate minority leader, Mitch McConnell of Kentucky, called it “laughable” as he spoke on the Senate floor on Thursday about the administration not being prepared to handle the expected increase in migrant crossings.

During the hearings this week, Democrats cited the plan and gave Mr. Mayorkas the opportunity to discuss it during testimony.

Republicans and Democrats sparred over which administration had a worse immigration policy, pointing to Mr. Trump’s restrictive approach and President Biden’s more liberal one.

At one point during the Homeland Security Committee hearing, Representative Elissa Slotkin, Democrat of Michigan, put Republicans, her fellow Democratic lawmakers and Mr. Mayorkas on notice.

“Our immigration system is broken,” she said. “Every single person here, along with the secretary, owns part of that responsibility.”

She added, “If you’re going to criticize what’s going on at the border, then put forward a piece of legislation to make it better. Don’t just use it as a political cudgel.”

Republicans and Democrats have proposed multiple pieces of legislation to address the border situation, but none have been able to break through the partisan divide.