The White House says it won’t let its plans to bring Syrian refugees to the U.S. be derailed by Friday’s terrorist attack in Paris, with a top official insisting Sunday that American authorities know how to weed out potential problems within the refugee community.

French President Francois Hollande has called the attacks, which killed at least 129 people and wounded more than 350 others, an act of war, and began retaliatory strikes on Islamic State strongholds in Raqqa, Syria.

Mr. Obama was in Turkey for a meeting of the Group of 20 nations, where the attacks were the top order of business.

Deputy National Security Adviser Ben Rhodes said the administration has a complete plan to both contain the Islamic State overseas and keep its recruited fighters out of the U.S. in a way that France was unable to.

But top members of Congress are skeptical of those plans, pointing to “gaping holes” in American defenses, and said even top Homeland Security officials have admitted the U.S. does not have access to the kinds of records and databases in the Middle East that would make sure immigration officers could screen out terrorists.

Refugees are the most pointed issue, but far from the only one, for American policymakers grappling with Friday’s devastating terrorist attack in Paris and its implications here at home.

Republican presidential candidates over the weekend demanded President Obama come up with a war plan to confront the Islamic State in Iraq and Syria. Former Arkansas Gov. Mike Huckabee went the furthest, saying the U.S. should refuse entry to anyone from a country where al Qaeda and the Islamic State have a significant presence.

Democratic presidential hopefuls, meanwhile, struggled with the issue over the weekend, trying to navigate between supporting Mr. Obama’s hazy strategy and coming up with their own ways of promising action on what’s become a global threat.

Officials said two of the attackers in Paris are now believed to have gained entry to Europe through a wave of refugees fleeing Syria. One of them appears to have come as part of a refugee surge in October in Greece, where he had no identification and was issued a Syrian passport, which he then used to travel through Europe, reaching France.

Hundreds of thousands of refugees are making similar journeys, overwhelming immigration officials tasked with screening them.

But Mr. Rhodes, speaking on “Fox News Sunday,” insisted Mr. Obama is sticking by his promise that the U.S. will take some 10,000 refugees in 2016, and even more in 2017.

“We have very robust vetting procedures for those refugees. It involves our intelligence community, our national counterterrorism center, extensive interviews, vetting them against all the available information,” Mr. Rhodes said.

The Obama administration assurances rang hollow on Capitol Hill, where the president has repeatedly underestimated the reach and desire of the Islamic State. He famously called it the “JV” team of terrorists just as it was beginning to advance in the Middle East, gaining a huge amount of territory and declaring a caliphate. And last week he said the movement was “contained” — just ahead of the Paris attack.

Members of Congress said Mr. Rhodes was dead wrong, and pointed to fears among Homeland Security officials who say they don’t, in fact, have access to the kinds of checks back in Syria that would allow them to vet would-be refugees.

Without access to those databases, and without people on the ground who can walk neighborhoods and verify details, there is no way to back-check a refugee’s story to see whether he is who he says he is. That puts enormous pressure on the in-person interview, conducted by Homeland Security officers.

Mr. Obama’s lieutenants say they believe they learned enough lessons from vetting refugees from Iraq and Afghanistan to get the Syria situation right — though in both of those cases they did have access to on-the-ground records.

“You can’t pick up the phone and call Syria, and that’s one of the reason I’ve said we won’t be able to take more refugees,” Sen. Marco Rubio, one of the GOP’s presidential candidates, said on ABC’s “This Week” program. “It’s not that we don’t want to, it’s that we can’t, because there’s no way to background-check someone that’s coming from Syria. Who do you call and do a background check on them?”

One of Mr. Rubio’s rivals, Louisiana Gov. Bobby Jindal, noted at the weekend that Syrian refugees already are being placed in his home state, and demanded that the Obama administration keep him informed following the terrorist attacks in Paris.

“As with former immigration crises and federal relocation policy, Louisiana has been kept in the dark about those seeking refuge in the state,” Mr. Jindal wrote, urging that Mr. Obama “pause” the entire relocation effort and warning that it could threaten Americans’ historic support for taking in refugees and other types of immigrants.

Several lawmakers on Capitol Hill are calling on their colleagues to include language in the upcoming spending bill that would constrain the president, preventing him from moving ahead with his Syrian refugee plans unless his top Homeland Security officials certify they believe they can accurately vet applicants.

Others say the bill should be used to block Mr. Obama altogether.

“The barbaric attacks in Paris should compel all western nations to acknowledge the reality of threat posed by radical Islam and to redouble their efforts to protect against it,” Sen. Jeff Sessions, chairman of the Senate’s immigration subcommittee, said in a statement to The Washington Times. “This should include action by Congress to block the president from unilaterally expanding refugee resettlement in the upcoming government funding bill.”

Republicans pressed Mr. Obama to come up with a better strategy for combating the Islamic State, saying he’s done the bare minimum so far.

But Congress itself has failed to act. After demanding Mr. Obama submit a proposal for using military force — something he eventually did — Republicans have failed to take it up, leaving the president to fight the war as he sees fit.

Among Democrats, the issue is complicated by the ongoing Democratic presidential contest. On Saturday, at the second Democratic debate of the primary season, held in Des Moines, Iowa, Sen. Bernard Sanders blamed climate change for spreading terrorism, saying that global warming is spawning more droughts, which is pushing more people toward radical ideologies.

Former Secretary of State Hillary Rodham Clinton found herself on the defensive over how she and Mr. Obama missed the rise of the Islamic State. She said the blame lies with Iraq and with Syrian President Bashar Assad.

But all of the Democratic candidates echoed Mr. Obama in refusing to say the U.S. is fighting “radical Islam,” though Mrs. Clinton did say the U.S. was battling “jihadists.”

• S.A. Miller, reporting from Des Moines, Iowa, and Seth McLaughlin contributed to this article.

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