Accounts that U.S. border officials are facing a surge of Mexican immigrants claiming asylum by using a few key words prompted a top Republican senator to call for the pending immigration bill to be put on hold "until these abuses are ended."

Sen. Jeff Sessions, R-Ala., one of the most vocal critics of the comprehensive immigration overhaul being pushed on Capitol Hill, said the surge in asylum requests "has exposed another grave flaw" in the implementation of federal immigration law.

Documents obtained by Fox News show Immigration and Customs Enforcement has been paying for hotel rooms for dozens of recently arrived families to relieve overcrowding at processing centers in the San Diego area. Some ICE employees are working overtime and others have been asked to volunteer to work weekend shifts.

Sessions pointed to the wave of asylum requests in arguing that, while the Senate has already passed an immigration bill, the House should not proceed until the issue is addressed.

"The House cannot allow such actions to continue," Sessions told FoxNews.com in a written statement. "They must use every power they have to end this absurdity. No immigration bill should advance until these abuses are ended."

Sessions, as well as the unions representing some immigration officials, have made similar arguments before about other alleged enforcement gaps -- like the practice of not deporting illegal immigrants accused of low-level offenses. They claim the bill, as drafted in the Senate, does not address these gaps.

Supporters of a comprehensive immigration bill claim it would be a boon for the country on several levels. President Obama, asked about the bill during his press conference last Friday, said the economy would be "a trillion dollars stronger" if the bill is passed, allowing millions of illegal immigrants to apply for legal status and work on the books. He also pointed to boosted border security measures.

"We know that the Senate bill strengthens border security, puts unprecedented resources on top of the unprecedented resources I've already put into border security," Obama said.

The president said that if the bill makes its way to the House floor, "it would pass." House leaders, though, have signaled interest in taking up a series of smaller bills instead.

The reports about a surge in asylum requests have raised new concerns, amid claims that illegal immigrants have learned they can attempt to get asylum by using a few key words -- namely, by claiming they have a "credible fear" of drug cartels.

Julie Myers Wood, a former assistant secretary for the Department of Homeland Security under the George W. Bush administration, told Fox News the requests are "suspicious."

"While we welcome the U.S. being a place where refugees and (asylum seekers) can come, it's troubling, and you have to think about asylum fraud in this instance," she said.

She noted that claiming "credible fear" is just a first step in the process. She said what's "concerning" is that if they are released from detention, "maybe they won't show up for their hearings" and go under the radar.

Myers Wood called for the administration to create a "task force" to look at the possibility of asylum fraud.

"This clearly has to have been orchestrated by somebody," said former U.S. Attorney for Southern California Peter Nunez. "It's beyond belief that dozens or hundreds or thousands of people would simultaneously decide that they should go to the U.S. and make this claim."

Sources say one day last week, 200 border-crossers came through the Otay Mesa Port of Entry claiming asylum, while as many as 550 overflowed inside the processing center there and in nearby San Ysidro.

Fox News spoke to four agencies responsible for the San Diego situation last week. All deferred to the Department of Homeland Security press office in Washington, D.C., which issued this statement:

"Credible fear determinations are dictated by longstanding statute, not an issuance of discretion. The USCIS officer must find that a 'significant possibility' exists that the individual may be found eligible for asylum or withholding or removal.

"If the credible fear threshold is met, the individual is placed into removal proceedings in Immigration Court. The final decision on asylum eligibility rests with an immigration judge."

A spokesman for Sen. Marco Rubio, R-Fla., one of the leading Republican advocates for an immigration bill, said Tuesday that the administration's "failures to enforce immigration laws and the resulting distrust in the federal government" are among the biggest obstacles to passing a bill.

But spokesman Alex Conant said the Florida senator is hoping to use such a package to "end the status quo of de facto amnesty."

FoxNews.com's Judson Berger and Fox News' William LaJeunesse contributed to this report.