President Obama’s plan to accept 10,000 Syrian refugees is under fire after Friday’s terrorist attack in Paris, with one GOP presidential candidate saying the U.S. should halt its plans as long as it’s not able to assure Americans it can weed out potential terrorists.

The Islamic State, which controls significant territory in Syria, has claimed credit for the coordinated attacks, which have left 129 dead and 352 wounded in Paris — and had earlier claimed credit for downing a Russian jet over Egypt.

The Associated Press said a Syrian passport found near the body of an attacker was linked to a man who entered the European Union through a Greek island last month.

Sen. Ted Cruz, a Texas Republican who’s on the rise in the presidential campaign, said those events show the Islamic State, also known by the acronym ISIS, views itself at war with the U.S., and the Syrian refugee program Mr. Obama announced is a risk.

“We need to immediately declare a halt to any plans to bring refugees that may have been infiltrated by ISIS to the United States,” he said in a statement. “We need to redouble our efforts to prevent ISIS agents from penetrating our nation by other means.”

Fellow candidate Mike Huckabee went further, calling for a halt in all admissions of those coming from countries where the Islamic State or al Qaeda have a “strong presence.”

“During the debate last week, I stated that we should not admit those claiming to be Syrian refugees and was condemned by the left for that position. I was right, and the events in Paris affirm that,” Mr. Huckabee, a former Arkansas governor, said in a statement. “Even the far left and politically correct government of France has closed its borders. It’s time for a president who will act to protect Americans, not just talk and protect the image of Islam.”

Mr. Obama has ordered the U.S. to prepare to accept 10,000 Syrian refugees in 2016, with more expected in 2017, as he said Americans must do their part to help give a safety valve to the hundreds of thousands fleeing the war-torn region.

But officials have acknowledged vetting Syrians will be difficult because the U.S. does not have access to Syrian records, and it’s tougher to verify would-be refugees’ stories and backgrounds.

Much will depend on the in-person interviews conducted by Homeland Security officers — an area where officials insist they’ve become expert, thanks to experience vetting Iraqi refugees.

Early news reports out of Paris said a Syrian passport was found at the site of one of the attacks. And reports said Poland has already decided to suspend its participation in the European Union refugee program in the wake of the attack.

But refugee program defenders took to social media to urge that countries not stray from their commitments to taking in Syrians.

Some of the defenders said the violence in Paris is what the refugees are fleeing, and makes it all the more imperative to keep the programs going.

In the U.S., however, Mr. Obama was already having a tough sell. Some Republican members of Congress have proposed place restrictions on the refugee program as a condition in the major spending bill that’s expected to advance through Congress over the next month.

Sen. Charles E. Grassley, chairman of the Senate Judiciary Committee, said Congress should require the FBI and other counterterrorism officials certify their ability to vet refugees before the program goes into effect.

Spokespersons for Mr. Grassley and several other high-profile critics of the Syrian refugee program said Saturday it was too early to tell what effect the Paris attack would have on the debate.

• Seth McLaughlin contributed to this article.

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