Around 3 in 10 families apprehended for illegally entering the United States from Mexico cut off their ankle monitoring devices shortly after being released from Immigration and Customs Enforcement custody and told to show up for asylum hearings, an ICE official said Tuesday.

"Nearly 3 in 10 family units are cutting off their ankle bracelets at the beginning of the process when they've been released from our custody within days or weeks so they're not even going to get to that point where they could get the final removal order," Matthew Albence, executive associate director for ICE, testified before the Senate Homeland Security and Governmental Affairs Committee.

Albence said the Homeland Security Department's inability to get all illegal immigrants released from custody and told to appear for court hearings was in part due to people ditching their ankle bracelets and disappearing into the interior of the country.

Those waiting to hear if their asylum cases are approved or denied were more likely to keep their monitors in tact and show up for court, but once their cases were denied, they were not likely to show up when a judge would then later order them deported.

"They will comply up until the benefit of complying is not there," said Albence.

The committee's ranking Democrat, Sen. Claire McCaskill of Missouri, asked Albence how it was monitoring illegal immigrants whose asylum requests were denied and they were to be deported.

Albence said the ankle monitoring program, known as Alternatives to Detention, was being used at much lower rates among those who applied for asylum, were denied, and ordered deported. McCaskill was upset ICE focused more on monitoring asylum applicants than those who posed a threat by not showing up to court and broke the law by evading deportation orders.

[More: Ankle monitors lead to fewer deportations than detention: ICE]

"It reminds me of how we got people to court when they were charged with a crime. And we would never dream of after the jury had found them guilty and they were sentenced, we would never dream of deciding that would be the least intensive time of monitoring," McCaskill said. "We ought to figure out a way to make court appearances more likely."

Those whose asylum claims are denied may appeal the ruling within 30 days. McCaskill said "they could be monitored" for that time to ensure they are picked up for removal if the appeal fails.

But Albence said his agency lacks the money, personnel, and resources to actually go after noncriminal offenders who have been ordered deported.

"ICE has not been given resources to go out and effectuate these at-large arrests in many years," he said.