Protesters demand that President Obama protect the rights of unaccompanied minors in front of The White House on Thursday. (Yue Wu/The Washington Post)

House Speaker John A. Boehner (R-Ohio) raised doubts Thursday that Congress will be able to fulfill President Obama’s funding request to address the influx of illegal migrants crossing the U.S.-Mexico border before lawmakers leave Washington for their summer recess in two weeks.

His comments came amid a sharpening debate between Democrats and Republicans about whether economic conditions in Central America or U.S. immigration policy is more responsible for the increase in illegal border crossings in recent years. Several GOP lawmakers introduced more legislation Thursday in response to the crisis. Democrats quickly dismissed the GOP proposals as political posturing.

Asked about the prospect of approving Obama’s $3.7 billion request before a five-week break begins Aug. 1, Boehner said, “I would certainly hope so, but I don’t have as much optimism as I would like to have.”

Boehner said he is less optimistic because “there’s just been some comments made by our colleagues across the aisle that are going to make this much more difficult to deal with.”

That was a reference to growing opposition among Democrats to change a 2008 law that grants extra protections to youths from Central American countries who cross the U.S.-Mexico border. Republicans and some Democrats have said it will be necessary to tweak the law in order to fulfill Obama’s request for emergency funding.

Public chastises both Obama and Republicans on immigration wave

Boehner said during his weekly news conference that the law must be changed if Obama expects to receive additional funds, because “it’s being abused.”

Other Republicans said that recent steps taken by Obama are directly responsible for the surge.

Sen. Ted Cruz (R-Tex.) said he planned to introduce a measure that would block the Obama administration from expanding a program launched in 2012 that allowed some children of illegal immigrants to stay in the country and obtain work permits or the deferral of deportation proceedings. Aides said that Cruz’s proposal would only block the expansion of the Deferred Action for Childhood Arrivals program, and that it does nothing to address the status of those already benefiting from the program.

Cruz told reporters that he rejected the idea that the 2008 law caused the flood of children at the border, saying that the surge happened only recently, after the 2012 executive action by Obama. “What caused this crisis was President Obama in 2012,” he said, suggesting that the “promise of amnesty” for children has led Central American families to send their children to the U.S. border.

“Then the 2008 law had an unintended consequence,” he said, forbidding border patrol agents from immediately returning those children from Central America.

His legislation would rescind Obama’s 2012 order, but he acknowledged that it did not deal with the children already here.

Sen. Lindsey O. Graham (R-S.C.) suggested that Cruz’s proposal “would help send the right signal. Boehner’s got to get something out of the House. No Republican’s going to write a check without changes­ to the law. How many changes in the law you get is the subject of negotiation. Democrats are getting hurt over this. We can go too far ourselves.”

But Senate Majority Leader Harry M. Reid (D-Nev.) blasted Cruz’s proposal, saying that “instead of considering a thoughtful, compassionate solution to a real-life crisis on our border, radical Republicans are trying to hold these kids ransom.”

Meanwhile, Sens. John McCain (R-Ariz.) and Jeff Flake (R-Ariz.) formally unveiled a plan that they said would make it less attractive for would-be immigrants to cross into the United States. In addition to changing the 2008 law, their plan would require that young migrants apprehended at the border be held in federal custody until they are repatriated or officials determine that their status allows them to remain in the country. Currently, the Department of Health and Human Services is required to reunite young illegal migrants with relatives if they have family in the country. McCain and Flake also call for temporarily hiring of about 300 personnel to help address a backlog of immigration cases and for increasing the number of refugee applications by up to 5,000 each for El Salvador, Guatemala and Honduras. But the three countries would be at risk of losing U.S. aid if the Obama administration cannot certify that they are taking steps to reduce the exodus.

Rep. John Carter (Tex.) and other House Republicans unveiled a similar proposal Thursday.

During a Senate Foreign Relations Committee hearing called to learn more about the crisis, administration officials said that more than 50,000 unaccompanied children from Central American countries have been apprehended along the U.S.-Mexico border this year. Nearly three-quarters are males between the ages of 15 and 17, the officials said.

Despite suggestions from Republican senators that Obama administration policies have inspired Central Americans to illegally travel north, officials repeatedly cited drug-related violence fueling economic instability in Central America.

Thomas A. Shannon, a State Department official with oversight of the border crisis, said that the political debate over immigration “does not have an impact” on the recent influx. And Bruce Swartz, a deputy assistant attorney general responsible for prosecuting drug and immigration crimes, said that Central American drug cartels and human smugglers “are marketing misunderstandings about how U.S. immigration law will work.”

As he sought information on conditions along popular immigration routes, Sen. Marco Rubio (R-Fla.) asked Shannon to confirm that local officials and human smugglers are advising women to use birth control or other contraception as they travel north due to the risk of rape.

“Not just women, but girls,” Shannon said.

Wesley Lowery and Paul Kane contributed to this report.