WASHINGTON – President Donald Trump on Saturday blamed Democrats for the high-profile deaths of two Guatemalan children this month while they were held in U.S. custody.

"Any deaths of children or others at the Border are strictly the fault of the Democrats and their pathetic immigration policies that allow people to make the long trek thinking they can enter our country illegally," Trump posted to Twitter on Saturday.

"If we had a Wall, they wouldn’t even try!" he added.

The president's comments come days after an 8-year-old boy from Guatemala died on Christmas Eve while in government custody, the second immigrant child to die in detention this month. A 7-year-old girl, also Guatemalan, died earlier in December in El Paso after being apprehended.

More:Guatemalan boy in US custody was diagnosed with a cold. Hours later, he died.

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Trump and congressional Democrats are locked in a furious dispute over funding for the president's proposed U.S.-Mexico border wall that has led to a partial government shutdown with no end in sight.

Trump said the children who died were “very sick” before they were placed in Border Patrol custody.

The New Mexico Office of the Medical Investigator said this week that the boy, Felipe Gomez Alonzo, tested positive for influenza B. Officials cautioned that the cause of death is still under investigation.

Felipe, along with his father, had been detained for a week after trying to cross the border illegally near El Paso on Dec. 18. Because of "capacity levels" in El Paso, they were moved to the Border Patrol station at Alamogordo, New Mexico, two days later, the CBP said. The next day, Felipe was sent to a hospital after a border agent noticed Felipe was coughing and had “glossy eyes,” the CBP said.

After being diagnosed with a cold and a fever, Felipe was prescribed amoxicillin and Ibuprofen. The CBP said he was held 90 minutes for observation and released Monday afternoon. That evening, he was sent back to the hospital with nausea and vomiting and died hours later.

Jakelin Caal died two days after a grueling trip through the desert with her father along with 161 other migrants who had crossed the New Mexico border illegally. They were initially taken to a base in rural New Mexico that did not have running water, according to Democrats who visited it after the girl’s death.

Jakelin and her father were scheduled to travel by bus to a Border Patrol station in New Mexico when her father, Nery Gilberto Caal, told Border Patrol agents she was sick. She was taken to a children's hospital in El Paso where she died.

Secretary of Homeland Security Kirstjen Nielsen traveled to El Paso on Friday to assess how the agency is conducting medical screenings and to review conditions at Border Patrol stations following the deaths. Nielsen called the deaths "deeply concerning and heartbreaking" and cited U.S. immigration system failings for a growing border crisis.

Nielsen said six people have died while in Border Patrol custody during Fiscal Year 2018, which ended in September, but that none were children.

Kevin McAleenan, commissioner of Customs and Border Protection, told "CBS This Morning" that the latest deaths are a "rare occurrence." "It's been more than a decade that we've had a child pass away anywhere in a CBP process so this is just devastating for us," he said.

Sen. Dianne Feinstein, D-Calif., wrote this week that the deaths are consistent with "previous reports of widespread abuse of children in immigration custody" and blamed them on facilities that "are not adequately staffed or equipped to properly care for children."

More:While the US debates caravans and wall, thousands of migrants are dumped in border cities

More:Two children have died in US border custody this month. Before that, none in a decade. Why now?

Allegations from families apprehended by Border Patrol agents, including that children were told they could drink water from a sink, but "are not given any cups" nor soap to wash their hands, were part of a raft of legal filings in August 2018.

Concerns about migrant children becoming sick – and the lack of medical care for them in ill-equipped Border Patrol stations – are far from new.

More:Two migrant children recently died in Border Patrol custody. There were documented warning signs

They had already been documented, in the same court case that establishes standards for how federal officials hold and release immigrant children.

Federal officials did not comment on the filings directly, but in an interview with The Arizona Republic, defended their handling of migrants and said border agents were not expected to be medical professionals.

The risk of illness for migrant children in custody, though, has been obvious to many who observed the system.

"As pediatricians, we say these detention centers are bad," said Dr. Colleen Kraft, the president of the American Academy of Pediatrics, which is now consulting with Homeland Security on providing better pediatric care. "They're cold, the lights are on 24/7, there are open toilets, and as a child, if you're not sick you can get sick."

Contributing: Pamela Ren Larson, Arizona Republic