U.S. Customs and Border Protection (CBP) Commissioner Kevin McAleenan said Sunday that while the recent sudden deaths of two children in the agency's custody were "absolutely devastating for us on every level," federal authorities had consistently followed protocol and were not to blame.

McAleenan told ABC's "This Week" that "our agents did everything they could as soon as these children manifested symptoms of illness to save their lives."

Jakelin Caal, 7, and 8-year-old Felipe Gomez Alonzo were each detained after entering the U.S. with their undocumented fathers. Jakelin suffered from dehydration and shock, while Felipe died just before 11:50 p.m. on Christmas Eve after falling ill with flu-like symptoms.

Felipe's father, Agustin Gomez, had seized “the opportunity” to travel to the Mexico border in hopes of migrating to the U.S. after hearing a rumor, according to Felipe’s stepsister, Catarina Gomez Lucas.

"We heard rumors that they could pass (into the United States). They said they could pass with the children," Lucas told the Associated Press.

"We need a sober-minded, nonpartisan look at our immigration laws to really confront and grapple with the fact that children and families are coming into this cycle, that’s first and foremost," McAleenan said Sunday.

Separately, South Carolina Republican Sen. Lindsey Graham told CNN's "State of the Union" that he would push for hearings to probe the childrens' deaths, as well as the reasons they were brought across the the border. Last week, California Democratic Sen. Dianne Feinstein had called on Graham, a member of the Senate Judiciary Committee, to take that step.

"I’m going to hold hearings on the deaths of these two children and the policies that entice people to come," Graham said.

Fox News has learned that Graham is set to meet with President Trump soon at the White House to discuss Syria policy and other issues. Fox News has also learned that behind the scenes this weekend, several senators are discussing a new potential compromise to end the ongoing partial federal government shutdown.

One bipartisan proposal is to provide $5.7 billion in funding for the border wall, as well as a congressional reauthorization of the Obama-era Deferred Action for Childhood Arrivals (DACA) program for those brought to the U.S. illegally as children, along with some other immigration provisions. There has also been talk about a special allowance for some classes of Central American refugees to be granted a more robust asylum status.

But despite the compromise talks, partisan tension remains high in Washington. After news of Felipe's death, top Democrats rushed to seemingly blame the Trump administration, even as key details remained unclear.

"Heartbroken and sickened by this news," New Mexico Democratic Sen. Martin Heinrich wrote on Twitter. "I am urgently demanding more details, but the Trump administration must be held accountable for this child’s death and all the lives they have put in danger with their intentional chaos and disregard for human life."

"We need a sober-minded, nonpartisan look at our immigration laws." — CBP Commissioner Kevin McAleenan

McAleenan added: "We also need to invest in Central America. The State Department’s announcement of an unprecedented increase in aid, I think, is a tremendous step forwards."

President Trump has repeatedly threatened to cut aid to Central American countries he maintains are not doing enough to curb illegal migration. "Guatemala, Honduras and El Salvador were not able to do the job of stopping people from leaving their country and coming illegally to the U.S.," he wrote in October. "We will now begin cutting off, or substantially reducing, the massive foreign aid routinely given to them."

On Dec. 28, Trump renewed that pledge on Twitter: "Word is that a new Caravan is forming in Honduras and they are doing nothing about it. We will be cutting off all aid to these 3 countries -- taking advantage of U.S. for years!"

U.S. immigration authorities said Wednesday that they have done new medical checks on virtually every child being held by the Border Patrol following the deaths.

Homeland Security Secretary Kirstjen Nielsen said Wednesday afternoon that she has ordered “extraordinary protective measures” after Felipe's death, which occured in immigration custody.

Border authorities did not disclose the results of the health checks.

But Nielsen said in a statement to Fox News that she had asked the Centers for Disease Control and Prevention to investigate what she says is an “uptick in sick children crossing our borders.” She also wants the U.S. Coast Guard to examine the medical programs offered by U.S. Customs and Border Protection, the agency that detained Felipe and his father for a week.

Nielsen pledged that all children in the future will receive a “more thorough” medical screening after they are apprehended.

The U.S. government’s system for detaining migrants crossing the border is under strain, and the U.S. is seeing a sharp rise in families with children.

Nielsen on Wednesday blamed “a system that prevents parents who bring their children on a dangerous illegal journey from facing consequences for their actions.”

Her department argues that it must detain more people to discourage other Central American families from trying to migrate.

“As a result of bad judicial rulings from activist judges and inaction by Congress, we are seeing a flood of family units and unaccompanied alien children,” Nielsen’s statement continued.

Fox News' Chad Pergram, Katherine Lam and Frank Miles contributed to this report.