A bipartisan group of senators say they are ready to act on concerns that the U.S. practice of waiving visas for short-term visitors from 38 countries could allow terrorists relatively easy access to American soil.

Sens. Jeff Flake (R-Ariz.) and Dianne Feinstein (D-Calif.) will introduce legislation Thursday tightening the visa waiver program to make it more difficult for potential radicals to enter the United States by ending visa waivers for anyone who has traveled to Syria or Iraq in the past five years.

The perpetrators of Friday’s terror attacks who have thus far been identified were French nationals who, because of that, would have been eligible to enter the United States with only minimal prior screening.

“The problem is the European communities, which are generally all visa waiver communities,” Feinstein, the top Democrat on the Senate Select Committee on Intelligence, said Wednesday after exiting a closed-door briefing with homeland security officials. “Let’s say France has had 2,000 people leave to go and fight. They are visa waiver countries, so the people come back to France and then they [can] come into the United States. The bill we would propose would strictly limit that.”

The new focus on the visa waiver program, which covered 20 million business and leisure travelers in 2014, comes during a week when there has been intense congressional focus on U.S. refugee policy amid fears that radicals would enter the country by posing as refugees fleeing the civil war in Syria.

Those seeking visa waivers must fill out an online application that is checked against security databases, and applicants are routinely denied waivers. But securing a traditional visa involves a more rigorous process, usually involving an in-person interview at an American consulate abroad.

Reaction to suggestions of halting the trickle of Syrian refugees has become politically divisive in recent days, with Democrats and Republicans at odds over whether even a “pause” in necessary. The House will vote Thursday on a bill to toughen the certification process for refugees from Iraq and Syria; President Obama threatened Wednesday to veto it, saying it would harm innocent victims of the Syrian civil war while doing little to keep terrorists out.

Concern about the visa waiver program, on the other hand, has taken on a less partisan tone.

Sen. Richard J. Durbin of Illinois, the Senate’s second-ranking Democrat, said several of the Paris attacks were on no-fly lists but that others were not — and, theoretically, could have boarded planes to the United States with French passports and entered the country with a visa waiver.

“That is a vulnerability far greater than 70,000 thoroughly vetted refugees,” Durbin said.

Flake, a member of the Foreign Affairs Committee, struck a similar tone, calling visa waivers “much more of a concern, frankly, than refugees,” as did Foreign Affairs Chairman Bob Corker (R-Tenn.): “The visa waiver program potentially is the place where there’s greater gaps, possibly, than the refugee program itself.”

Sen. Christopher Murphy (D-Conn.) said he sensed “openness” to visa-waiver changes among the Obama administration officials who briefed senators Wednesday. Those briefers included Homeland Security Secretary Jeh Johnson.

“I think there’s bipartisan support that could be found for changes to the visa waiver program in a way it likely won’t be found on a prohibition on refugees from the Syrian conflict,” Murphy said.

The Department of Homeland Security made some changes to the visa waiver program in August in response to similar concerns about the threat from radicals who hold European passports. Countries who wish to participate in the program must use passports that include biometric data about their holders and share more data about travelers.

Feinstein said in January, after the Charlie Hebdo attacks in Paris, that she planned to introduce legislation aimed at tightening gaps in the program, but those plans did not advance.

Karoun Demirjian contributed to this report.