WASHINGTON — House Republicans on Thursday pushed ahead with a sweeping immigration crackdown that would codify several stringent border policies imposed by the Trump administration, after months of internal feuding that led G.O.P. leaders to drop some of the plan’s most extreme provisions.

The House Judiciary and Homeland Security Committees in recent days approved their pieces of the plan, which has little chance of being considered in the Democratic-led Senate but sets up a pivotal test of whether Republican leaders can deliver on their campaign promise to clamp down on record migrant inflows.

For Republicans, who have repeatedly attacked President Biden on his immigration policies and embarked on an effort to impeach his homeland security secretary, the measure is a chance to lay out an alternative vision on an issue that galvanizes its right-wing base.

The legislation, now expected on the floor next month, would direct the Biden administration to resume constructing the border wall that was former President Donald J. Trump’s signature project. It would also mandate that employers check workers’ legal status through an electronic system known as E-Verify and reinstate the “Remain in Mexico” policy, forcing asylum applicants to wait in detention facilities or outside the United States before their claims are heard.

The plan “will force the administration to enforce the law, secure the border, and reduce illegal immigration once again,” Representative Mark E. Green, Republican of Tennessee and the Homeland Security Committee’s chairman, said during the panel’s debate on Wednesday.

Democrats have derided the package as misguided and draconian, accusing Republicans of seeking to invigorate their core supporters in advance of the 2024 election by reviving some of Mr. Trump’s most severe border policies. They made vocal objections to provisions that would ban the use of the phone-based app known as “C.B.P. One” to streamline processing migrants at ports of entry, expedite the deportation of unaccompanied minors, and criminalize visa overstays of more than 10 days.

Republicans “want to appeal to their extreme MAGA friends more than they want progress,” Representative Bennie Thompson of Mississippi, the top Democrat on the Homeland Security Committee, said Wednesday, calling the Republican legislation a “profoundly immoral” piece of legislation that would “sow chaos at the border.”

Still, the package represents a compromise of sorts between hard-right Republicans and more mainstream G.O.P. lawmakers, including a mostly Latino group from border states that balked at proposals that threatened to gut the nation’s asylum system.

The party’s immigration plan — which top Republicans had hoped to pass as one of their first bills of their new House majority — has been stalled for months. A faction led by Representative Tony Gonzales, Republican of Texas, has raised concerns about the asylum changes, threatening to withhold votes that Speaker Kevin McCarthy, Republican of California, cannot afford to lose given his slim majority.

Over the last week, G.O.P. leaders have quietly made a series of concessions to win over the skeptics. Republicans on the Judiciary Committee agreed to drop a provision that would have effectively stopped the intake of asylum seekers if the government failed to detain or deport all migrants seeking to enter the country without permission. But the measure still contains a number of new asylum restrictions.

“It’s in a good spot,” Mr. Gonzales said of the legislation on Thursday, saying that the changes made to the asylum provision had satisfied his concerns. “As long as nobody does any funny business — you’ve got to watch it till the very end.”

G.O.P. leaders predicted on Thursday that they would be able to draw a majority for the legislation when it comes to the House in mid-May, a timeline selected to coincide with the expected expiration of a Covid-era policy allowing officials to swiftly expel migrants at the border. The termination of the program, known as Title 42, is expected to inspire a new surge of attempted border crossings and supercharge the already bitter partisan debate over immigration policy.

But it was unclear whether Republicans who had objected to the E-Verify requirement would be on board.

Representative Thomas Massie, Republican of Kentucky withheld his support for the Judiciary Committee’s bill because of the work authorization mandate, arguing that people “shouldn’t have to go through an E-Verify database to exercise your basic human right to trade labor for sustenance.”

Such databases “always get turned against us, and they’re never used for the purpose they were intended for,” added Mr. Massie, a conservative libertarian.

Representative Dan Newhouse, a Republican farmer in Washington State, has expressed concern that the E-Verify mandate could create labor shocks in the agricultural sector, which relies heavily on undocumented immigrant labor. Though the legislation delays the requirement for farmers for three years, Mr. Newhouse has argued that any such change should be paired with legislation creating more legal pathways for people to work in the United States.

With the expected floor vote just weeks away, G.O.P. leaders have been treading carefully, even making last-minute concessions to Democrats in hopes of bolstering support for the legislation.

During the wee hours on Thursday morning, as the Homeland Security Committee debated its bill, Republicans pared back language barring nongovernmental organizations that assist undocumented migrants from receiving funding from the Department of Homeland Security. They did so after Democrats pointed out the broadly phrased prohibition could deprive legal migrants and U.S. citizens of critical services as well.

Their changes did not go far enough to satisfy Democrats, who unanimously opposed the package on the Judiciary and the Homeland Security panels — and are expected to oppose the combined border security package en masse on the House floor.

They have also argued that any measure to enhance border security or enforcement must be paired with expanded legal pathways for immigrants to enter the United States.