President Obama threatened late Wednesday to veto legislation aimed at improving screening for Syrian refugees, potentially putting the White House and Congress on a collision course in a matter of days.

The veto threat came as the House was preparing the bill -- which sets high hurdles for refugee admission including FBI background checks and sign-offs by top officials -- for floor action as early as Thursday. In a committee meeting, Rep. Pete Sessions, R-Texas, accused the president of confusing the public about the intentions of the legislation.

Moments later, the White House issued a statement defending the current screening process and claiming the changes called for under the bill would create "significant delays and obstacles" for the existing vetting program.

"Given the lives at stake and the critical importance to our partners in the Middle East and Europe of American leadership in addressing the Syrian refugee crisis ... [Obama] would veto the bill," the White House said.

But House Republicans touted the legislation as a common-sense answer to security concerns.

Further, while Republicans a day earlier called for a "pause" in Syrian refugee admissions, some on Wednesday indicated a willingness to accept refugees from Syria and Iraq who are fleeing the civil war and Islamic State militants -- provided the screening process is improved, under the terms of the bill.

"America has a proud tradition of welcoming refugees into our country, and we lead the world in humanitarian assistance. However, we also must put proper measures in place to ensure our country's safety," House Homeland Security Committee Chairman Michael McCaul, R-Texas, said in a statement.

The bill introduced Wednesday would require the FBI director to certify a background investigation for each refugee -- and several top security officials to certify that each refugee is not a security threat to the U.S. -- before a refugee from Iraq or Syria can be admitted.

House Speaker Paul Ryan said the chamber would vote on the bill later this week, and stressed that it would not subject applicants to a religious test. He made this clarification after some GOP presidential candidates suggested preference should be given to Christians.

McCaul, in a statement, said that while he wants a "temporary suspension" of Syrian refugee admissions, "It is apparent that the President will ignore these concerns, making this legislation necessary to toughen security measures."

Indeed, Obama on Wednesday continued to defend plans to bring in an additional 10,000 Syrian refugees over the next year as he threatened to veto the House bill.

"Slamming the door in the face of refugees would betray our deepest values. That's not who we are. And it's not what we're going to do," Obama tweeted late Wednesday morning.

Rep. Richard Hudson, R-N.C., also touted the latest House bill, saying it was based on legislation he introduced just days earlier. Hudson said the new bill would likely be voted on in the House on Thursday.

"America is a compassionate nation. No country on Earth does more or spends more to care for our fellow man. But being compassionate doesn't mean we have to have reckless policies that put American lives at risk," he said in a statement.

McCaul's committee also released a report Wednesday on the Syrian refugee flow, saying it reveals "alarming gaps in the vetting of Syrian refugees at home and abroad."

Already, the new legislation was facing criticism from both sides of the aisle -- and not just the White House.

Heritage Action executive officer Michael A. Needham said in a statement that the bill, while setting up better vetting, "provides no leverage for Congress to weigh in and relies solely on President Obama's appointees to carry out the new vetting process."

House Democrats also voiced opposition, with one House Democratic leadership aide telling Fox News the bill would "end the refugee program altogether." The aide said they hope to "negotiate a bipartisan bill" and are weighing introducing an "alternative bill."

Meanwhile, CIA Director John Brennan said in a speech Wednesday that about half of Syria's population -- or about 12 million people -- has been displaced by the ISIS onslaught and the civil war, a number that includes both those who have been internally displaced and those forced to flee the country.

Speaking at the Overseas Security Advisory Council Conference, Brennan said Syria is "approaching 50 percent of the population" that has been displaced.

Fox News' Mike Emanuel and The Associated Press contributed to this report.