Tashfeen Malik, who with her husband carried out the massacre in San Bernardino, Calif., passed three background checks by American immigration officials as she moved to the United States from Pakistan. But none uncovered what Ms. Malik had made little effort to hide — that she talked openly on social media about her views on violent jihad.

She said she supported it. And she said she wanted to be a part of it.

American law enforcement officials said they recently discovered those old — and previously unreported — postings as they pieced together the lives of Ms. Malik and her husband, Syed Rizwan Farook, trying to understand how they pulled off the deadliest terrorist attack on American soil since Sept. 11, 2001.

Had the authorities found the posts years ago, they might have kept her out of the country. But immigration officials do not routinely review social media as part of their background checks, and there is a debate inside the Department of Homeland Security over whether it is even appropriate to do so.

The discovery of the old social media posts has exposed a significant — and perhaps inevitable — shortcoming in how foreigners are screened when they enter the United States, particularly as people everywhere disclose more about themselves online. Tens of millions of people are cleared each year to come to this country to work, visit or live. It is impossible to conduct an exhaustive investigation and scour the social media accounts of each of them, law enforcement officials say.

In the aftermath of terrorist attacks in San Bernardino and Paris, this screening process has been singled out as a major vulnerability in the nation’s defense against terrorism. Lawmakers from both parties have endorsed making it harder for people to enter the United States if they have recently been in Iraq or Syria. Donald J. Trump, the Republican presidential candidate, has said there should be a temporary ban on Muslims’ entering the country.

While President Obama has cautioned against “a betrayal of our values” in the way the United States responds to threats, he has ordered a review of the K-1 visa program, which allows foreigners like Ms. Malik to move to the United States to marry Americans, putting them on a pathway to permanent residence and, ultimately, citizenship.

The Obama administration is trying to determine whether those background checks can be expanded without causing major delays in the popular program. In an attempt to ensure they did not miss threats from men and women who entered the country the same way Ms. Malik did, immigration officials are also reviewing all of about 90,000 K-1 visas issued in the past two years and are considering a moratorium on new ones while they determine whether changes should be made.

“Somebody entered the United States through the K-1 visa program and proceeded to carry out an act of terrorism on American soil,” the White House spokesman, Josh Earnest, said on Thursday. “That program is at a minimum worth a very close look.”

In an era when technology has given intelligence agencies seemingly limitless ability to collect information on people, it may seem surprising that a Facebook or Twitter post could go unnoticed in a background screening. But the screenings are an example of the trade-offs that security officials make as they try to mitigate the threat of terrorism while keeping borders open for business and travel.

“We run people against watch lists and that’s how we decided if they get extra screening,” said C. Stewart Verdery Jr., a senior Homeland Security official during George W. Bush’s administration. “In cases where those lists don’t hit, there’s nothing that distinguishes them from people we would love to welcome to this country.”

Ms. Malik faced three extensive national security and criminal background screenings. First, Homeland Security officials checked her name against American law enforcement and national security databases. Then, her visa application went to the State Department, which checked her fingerprints against other databases. Finally, after coming to the United States and formally marrying Mr. Farook here, she applied for her green card and received another round of criminal and security checks.

Ms. Malik also had two in-person interviews, federal officials said, the first by a consular officer in Pakistan, and the second by an immigration officer in the United States when she applied for her green card.