Democratic Rep. Henry Cuellar and Republican Sen. Lindsey O. Graham are teaming up to urge the White House to name former homeland security secretary Jeh Johnson or someone with similar credentials to coordinate efforts to combat the influx of migrants to the southern border — a fresh signal of frustration with President Biden’s strategy among prominent lawmakers.

The border-district Democrat from Texas and the outspoken Republican from South Carolina made their call in a letter Friday, a move that reflects the turbulent politics of immigration and its political challenge for Biden. The surge in illegal crossings to the United States shows no sign of slowing despite the administration’s early predictions that it would fade as summer temperatures climbed.

Although Cuellar said the call is not a criticism of the personnel Biden has tapped to tackle immigration, including Vice President Harris, the letter amounts to a sharp critique of the administration’s approach and a request for the president to toughen his posture.

“We know this current influx is neither seasonal nor temporary,” the lawmakers say in their letter, addressed to Biden and Homeland Security Secretary Alejandro Mayorkas. Cuellar’s office shared the letter with The Washington Post.

Asserting that Homeland Security facilities and staffers “are overwhelmed,” and raising concerns about the spread of the coronavirus in areas near the border, Cuellar and Graham add that “the situation on the border will worsen unless we enact policies that will end the surge and fix our broken immigration system.”

The request that Biden name a border czar — which the lawmakers say should be done by creating a “special executive appointment” — is somewhat unusual, since members of Congress often complain when presidents choose such figures, who are not subject to Senate confirmation. The letter argues that Johnson “carries an exceptional bipartisan reputation for pragmatic approaches,” and urges the president to appoint “someone with the unique credentials of former Secretary Johnson to this position.”

The White House contends that it has made significant progress on the border, particularly in caring for unaccompanied minors entering the United States, and that the immigration system will remain troubled until Congress passes a bill to overhaul it. Yet Cuellar said he thinks a more immediate course correction is needed.

“Something has to change, because the border communities are having a very difficult time right now,” Cuellar said in an interview late Thursday. Immigrants now feel free to enter the United States, he added: “There is this feeling of a green light at the border.”

Johnson declined to comment. A White House spokesman did not have any immediate comment. A White House aide speaking on the condition of anonymity to discuss the situation said the White House had no new personnel announcements to share.

As for why Johnson, who served during President Barack Obama’s second term, would be a good choice, Cuellar said, “He was compassionate, but at the same time he did what had to be done, and he did not worry about offending the far left or the immigration activists.” He said he had not spoken to Johnson yet about the idea.

While Johnson is getting fresh bipartisan praise, he and Obama faced criticism during their tenure from Democratic activists who said they were too harsh on undocumented migrants, prompting many to derisively label Obama the “deporter in chief.” Anger over his immigration strategy has lingered, prompting Biden to make a rare break from Obama’s policies as he campaigned last year.

The administration has toughened its rhetoric on immigration, telling Latin Americans that the border is not open and they should stay home. At the same time, immigration activists inside and outside the administration continue to push for policies they say are humane and necessary after the harsh approach of the Trump years, presenting Biden with a tough balancing act.

The call to create a new position and empower a former Obama administration official could be seen as an implicit rebuke of Harris. Biden has designated the vice president as his point person on migration issues, and she traveled to the border last month.

Harris has been criticized both by liberals who think her rhetoric was too stern during a recent trip to Latin America and by Republicans who say she has been ineffective in slowing the flow of migrants.

Harris has responded that she is not Biden’s “border czar,” despite GOP efforts to label her that way, stressing that her mission is rather to address the dire conditions in Latin American countries that prompt its residents to leave in the first place. The administration Thursday issued what it called its “root causes strategy” on migration, outlining goals for Latin American countries on issues from poverty and corruption to oppression and violence.

In the interview Thursday, Cuellar denied that the call to bring in Johnson was a vote of no confidence in Harris. “This has no reflection on anybody that has a responsibility with the border,” Cuellar said. “Like anything else, you know, a business or a CEO, there’s nothing wrong when you bring an expert in. It’s smart for a leader.”

Still, the lawmakers’ letter underlines the persistent worries among many Democrats about the political and policy impact of Biden’s immigration strategy. Americans give Biden poor marks for his handling of the issue, polls show, and the president himself has worried about GOP attacks on these topics, according to people familiar with his thinking.

When it comes to border issues, Cuellar, who represents a district along the border with Mexico, has been one of Biden’s most vocal Democratic critics. His decision to ally himself with Graham — a onetime member of the “Gang of Eight” senators seeking a comprehensive deal on immigration, who later reinvented himself as a staunch Trump ally — is a reflection of the upside some Democrats increasingly see in distancing themselves from the president on immigration and crossing the aisle to create partnerships with Republicans.

The difficult calculations for Democrats about how and when to be critical of the president are especially stark in South Texas, where GOP gains in 2020 alarmed Democratic strategists. Cuellar and Rep. Vicente Gonzalez (Tex.), another border-district Democrat, have been blunt in expressing concerns about the administration’s strategy.

Resurgent fears about the spread of the coronavirus, meanwhile, have compounded an already difficult situation. An increase in infections in Mexico and along the southern border has stalled the administration’s plans to phase out a controversial public health order used to return many migrants to Mexico, The Post reported Wednesday.

Although Biden and his top aides repeatedly characterized the surge of migrants as seasonal, there were more than 188,000 illegal crossings in June, despite the summer heat. That total marked a 20-year high.

Republican strategists and officials have seized on these trends as part of their midterm campaign message that the country is descending into chaos under Biden and that he has made Americans less safe. Republicans have also pointed to a spike in homicides to bolster that case.

The Republican criticism and the poor polling trends have stoked concern among Democrats inside and outside the White House, according to people with knowledge of the situation. Strategists in both parties say it is still unclear which topics will dominate the midterm debate, but many on both sides see immigration as unfavorable terrain for Biden.