More than 2 million illegal immigrants will be approved for President Obama’s deportation amnesty over the next few years, and they will be eligible to collect Social Security and Medicare benefits as well as claim a special tax break for low-income families, the Congressional Budget Office said in an analysis Thursday.

Mr. Obama predicted that up to 5 million illegal immigrants could be eligible for his amnesties, but the CBO numbers predict only 2.25 million will have signed up and been approved by 2017.

The estimate was released as the administration defended the law in a federal court in Texas on Thursday, asking a judge to reject a request by Texas and two dozen other states to halt the program even before it gets started.

Judge Andrew S. Hanen, sitting in Brownsville, said he won’t rule before the end of the month. Applications for the first part of the amnesty are scheduled to begin in the middle of February.

“There aren’t any bad guys in this,” Judge Hanen told attorneys for both sides, according to The Brownsville Herald. He gave no indication of which way he is leaning in the thorny case, which is likely to determine Mr. Obama’s legacy on immigration.

Texas and its allies argue that Mr. Obama overstepped his legal bounds in November when he announced a program to halt deportations for illegal immigrant parents who have legal resident or U.S. citizen children, and to expand a 2012 amnesty for illegal immigrants who were brought to the U.S. as children.

To win, the states first must prove that they were injured by the amnesty, which would give them “standing” to sue. Then they must prove that Mr. Obama’s actions are either unconstitutional because they try to rewrite the laws, which is Congress’ job, or else they are official policies that should have been submitted to the public for comment and revisions before they were enacted.

Administration attorneys told Judge Hanen that Mr. Obama isn’t rewriting law, but rather deciding whom to prosecute under it based on his powers of prosecutorial discretion. The attorneys say presidents going back to the 1950s have used similar powers to halt deportations, albeit on smaller scales.

The Obama administration said if it declares most illegal immigrants off limits for deportation, it will be easier to pursue the recent illegal immigrants and the serious felons who won’t qualify for the leniency.

Mexican officials, hoping to help their citizens stay in the U.S., began issuing birth certificates at consulates Thursday.

Mexicans make up the majority of the illegal immigrant population in the U.S., though Central Americans may be rivaling them among newcomers, according to statistics.

Although Judge Hanen didn’t tip his hand about his thinking, administration supporters fear the worst. They point to a striking order he issued in December 2013, at the beginning of the surge of Central American children crossing the border, that accused the Homeland Security Department of being complicit in human trafficking.

Judge Hanen said because agents would take children caught at the border to their parents living illegally in the U.S., without trying to deport either of them, the government was in effect doing the job of smugglers and encouraging others to make the same journey.

“Clearly, the plaintiffs filed their suit in Brownsville for one reason — a friendly judge,” said America’s Voice Education Fund, which lobbies for immigrants’ rights.

Three self-identified illegal immigrants, who filed as “Jane Doe” litigants, asked Judge Hanen on Thursday to be allowed to join the lawsuit in defense of the president’s policies.

The Mexican American Legal Defense and Education Fund, which filed on their behalf, said they “could face deportation and separation from their families and their communities” if the president’s amnesty is struck down.

They are three of the millions who could qualify, but the CBO report suggested that many of those who are eligible for amnesty won’t apply.

All told, the budget analysts predicted that 1.5 million will have been approved under the amnesty for illegal immigrant parents by 2017, while 750,000 illegal immigrants will have been approved under the modified 2012 amnesty for Dreamers, who are illegal immigrants brought to the U.S. as children.

The total illegal immigrant population in the U.S. is estimated to be 11 million to 12 million people. The number dropped at the beginning of the Obama administration but has been ticking up in recent years.

Most illegal immigrants who do not qualify for the amnesty are unlikely to be deported under the guidelines Mr. Obama issued in November, which call for only the most serious criminals to be deported.

On Wednesday, the House voted to insist that convicted sex criminals be included in that list of crimes serious enough to demand deportation.

Those who are granted amnesty are also entitled to work permits, allowing them to compete legally for jobs. Texas and its allied states argued that those who get the amnesty are also eligible for some state benefits and services, such as a driver’s license and, in some states, concealed weapons permits or health care.

The CBO also said those covered by the temporary amnesty, which the government refers to as “deferred action,” will be eligible for some federal benefits.

“Because they are lawfully present during the period of their deferred status, they are eligible to receive Medicare and Social Security benefits if they meet the programs’ requirements,” the CBO said in its report. “In addition, those individuals who are approved for deferred action and receive work authorization have Social Security numbers and therefore can claim the earned income tax credit if they qualify. They are ineligible for other federal benefit programs.”

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