President Barack Obama, who’s bound to be hammered no matter what action he takes on the Texas border crisis, tried to reassure skeptical allies Wednesday, telling members of the Congressional Hispanic Caucus that Central American migrants’ rights to due process would be honored even as he’s pushing to accelerate the process for deporting them.  

Obama faced a caucus almost unanimously opposed to changing a 2008 human trafficking law granting children rights to see a judge and contest deportation — and they left the meeting sounding happy Obama had at least heard their concerns. “We told the President the CHC will not support a supplemental budget request that undermines the legal protection for children and we will work with him to get the resources our government needs to respond to the humanitarian crisis we are seeing with child refugees from Central America,” Rep. Luis V. Gutiérrez, D-Ill., said in a statement. “The CHC told the President we will fight for the rights of these children under the law and a lot of other Democrats will stand with us.”  

Lawmakers told reporters that they do not oppose expediting the hearings for the children, but said they should not lose their rights to see a judge.  

In an interview, Gutiérrez predicted only a small fraction — 5 percent to 10 percent — would ultimately qualify to stay in the United States and the rest would be deported. But he said the message to Central American parents is they should not send their children here, that this is the wrong way to come to the United States.  

Separately, the CHC pushed for Obama to end deportations for millions of long-term, unauthorized immigrants, in particular deportations that would split families, many of whom have members who are legally in the United States alongside those who are not.  

Obama cautioned the lawmakers that he would be constrained by what his lawyers tell him he is authorized to do under the law, but would do everything he could, Gutiérrez said.  

Gutiérrez said the president agreed to set up a meeting in the next two days between the CHC and the legal counsel at the Department of Homeland Security.  

While Obama and Vice President Joseph R. Biden Jr. sought to win over the CHC Wednesday afternoon, Homeland Security Secretary Jeh Johnson briefed senators at the Capitol about the president’s $3.7 billion border request. Senators in both parties said Johnson indicated he supported changing the 2008 law and expressed urgency about the need for more money from Congress.  

“If we don’t come up with a supplemental appropriation, the Department of Homeland Security says they will run out of money by the middle of August, and the Department of Health and Human Services faces similar deadlines,” said Senate Majority Whip Richard J. Durbin, D-Ill.  

Asked how he thought this would play out among Democrats opposed to significant changes to the 2008 law, Republican Lindsey Graham of South Carolina said he anticipated “that they will fold. They will fold like a cheap suit.”  

Niels Lesniewski contributed to this report.