There is growing evidence that a surge of tens of thousands of Central American minors across the Mexican border into Texas is being driven in large part by the perception they will be allowed to stay under the Obama administration’s immigration policies.

Administration officials — after initially dismissing such reports — are now attempting to push back on the idea, warning parents not to send their children as officials scramble to accommodate tens of thousands who already have arrived in Texas.

“Those who cross our border today illegally, even children, are not eligible for an earned path to citizenship,” Homeland Security Secretary Jeh Johnson said this week. “Those apprehended at our borders are priorities for removal . . . regardless of age.”

The crisis marks another immigration-related political dilemma for President Obama and has been seized on by Republicans as an argument against legislation to allow millions of undocumented immigrants to remain in the country legally. It could also hamper Obama’s ability to meet the demands of his liberal base by using executive authority to ease enforcement policies if Congress fails to act.

The administration has emphasized that the influx of minors is being driven foremost by widespread gang-related violence in Guatemala, Honduras and El Salvador. Most of the children are crossing over the Rio Grande Valley into Texas, often under the guidance of human-smuggling cartels, and many are seeking to reunite with parents already in the United States, aid workers said.

The number of minors apprehended in the past nine months who came from those three countries alone is 34,611 — 31 / 2 times more than all of 2012. That was the year Obama announced that he would defer the deportations of young immigrants brought to the country illegally by their parents before 2007.

An internal memo from the U.S. Border Patrol last month estimated that, all told, 90,000 minors would be apprehended this year and 142,000 next year.

Republicans point to the crisis as evidence that Obama’s policies have contributed to a widespread belief that young migrants will be allowed to remain in the country. Sen. Dianne Feinstein (D-Calif.) said Thursday that her staff found evidence on a recent tour of a Border Patrol station in Nogales, Ariz., that the young migrants are being promised as much before leaving home.

Two of Feinstein’s staff members visited the Nogales facility, where hundreds of children were sent for processing in recent weeks after the patrol stations in the Rio Grande were overwhelmed. In a statement to The Washington Post, Feinstein said the children were being well fed and cared for, but she emphasized that “what concerns me is why the children are crossing the border in the first place.”

“After engaging with the children and U.S. personnel, my staff learned that many of the children were smuggled across the border after hearing radio ads promising they would not be deported,” Feinstein said. “My staff also heard that religious organizations are spreading the same message.”

A leaked internal memo written by Border Patrol agents in the Rio Grande Valley offers additional evidence of such perceptions. The document was distributed to reporters and congressional offices this week by the conservative Center for Immigration Studies, which advocates for limited immigration. It is a summary of interviews on May 28 with 230 youths and women from Honduras, Guatemala and El Salvador who were apprehended trying to enter the United States.

The main reason the migrants had crossed into the United States was “to take advantage of the ‘new’ U.S. law that grants a free pass or permit” from the government, referred to in their home countries as “permisos,” the memo stated.

Gang-related violence, economic hardship and domestic abuse also were cited as reasons for the exodus from Central America. But “the subjects also indicated that ‘everyone’ in their home countries is aware that ‘permisos’ are being issued to family units in south Texas,” according to the memo. “The news of these ‘permisos’ is spread by word of mouth and international and local media.”

At a Senate Judiciary Committee hearing this week, Johnson told lawmakers that he had not seen the Border Patrol memo and did not agree with its conclusions. Officials at the Department of Homeland Security did not respond to several requests from The Post to discuss the document.

Cecilia Muñoz, the White House’s director of domestic policy, acknowledged last week that unfounded “rumors” could be playing a role in the influx of minors. But she and Johnson have rejected the notion that such perceptions are a leading cause of the crisis.

One Senate Democratic aide who has worked closely on the issue suggested that human-smuggling networks in Central America are falsely propagating the rumors to recruit clients, whose families pay large sums for the trip north.

With immigration reform stalled on Capitol Hill, Obama has been under pressure from liberals to expand the deferral program he announced for young migrants in 2012, and the president has instructed Johnson to oversee a review of enforcement policies. Obama delayed the review until after the summer in hopes of persuading House Republicans to support a legislative solution.

GOP lawmakers warned that the president could make the problem worse if he loosens restrictions further on his own.

“It’s been widely reported that the administration is contemplating yet another amnesty . . . like two years ago,” Sen. Ted Cruz (R-Tex.) said during the committee hearing. “Granting another amnesty will result in those numbers being even higher, more little girls and little boys subjected to violence and horrific, dangerous conditions. It’s a serious mistake to go down that road.”

Democrats and immigration advocates say Republicans are exploiting the situation for political gain and point to Immigration and Customs Enforcement data showing the Obama administration has deported more than 2 million migrants.

Johnson, who plans to travel to Guatemala in July, said the administration is launching a television, radio and newspaper campaign in that country and others warning parents there of the dangers of putting their children in the hands of smuggling cartels.

Even some Republicans who favor immigration reform have argued that the administration has not been clear enough. Sens. Jeff Flake and John McCain, Arizona Republicans who were among a bipartisan group of eight senators who drafted a Senate immigration bill last year, sent Obama a letter this week asking him to make a public statement that children who cross the border will be deported.

Like Cruz, Flake said it would be a mistake for Obama to move forward on executive actions to ease deportation policies in light of the current crisis.

“The president is under pressure from his own base on deportations, and if he were to stand and say, ‘We will deport you’ — that is a difficult statement to make with his own base,” Flake said. “But, man alive, it’s a statement he should be making.”