About 2.5 million illegal immigrants have settled in the U.S. during President Obama’s tenure, according to estimates being released Monday by the Center for Immigration Studies, which said it’s an improvement compared with the Bush administration.

Nearly 800,000 of those illegal immigrants arrived in the past two years, suggesting that the flow has ticked up as the economy has improved and as Mr. Obama has reshaped enforcement policies, focusing on criminals while relaxing actions against rank-and-file illegal immigrants.

Still, the total illegal immigrant population has remained steady at an estimated 11 million to 12 million over the past six years, the report concluded, finding that the arrivals are canceled out by the hundreds of thousands who return home, die or earn legal status through existing channels such as marrying an American.

Steven A. Camarota, research director at the center and the author of the study, said 1.5 million to 1.7 million illegal immigrants arrived from 2009 through 2013, and he calculates, based on Census Bureau data, that an additional 790,000 came between the middle of 2013 and May of this year.

While not a major surge, and less than the annual increases of 500,000 to 600,000 during President George W. Bush’s term, it suggests a porous border and visa system. Indeed, Mr. Camarota said it’s likely that the illegal flow increasingly isn’t people crossing the land border with Mexico, but rather entering legally and then overstaying visas.

He said that means the newest illegal immigrants are less likely to be Mexican and more likely to be from Europe, Asia and Africa.

“You can see it as a missed opportunity. In a way, illegal immigration naturally falls as a result of out-migration, deaths and legalization. But we didn’t stop the mass flow coming in. We just replaced all those folks,” he said.

The immigration debate is heating up again as a result of a killing in San Francisco attributed to that city’s sanctuary policy and Mr. Obama’s legal battle to grant tentative legal status and work permits to yet more illegal immigrants.

The San Francisco killing has refocused the debate on enforcement, particularly with the revelation that the suspect had been deported five times and managed to sneak back into the U.S. each time.

Republican presidential candidate Donald Trump has fanned the debate, accusing Mexico of sending some of its society’s worst elements to the U.S.

Democrats say Mr. Trump has gone overboard and that enforcement is up to par.

“The border has never been more secure than it is now,” Rep. Joaquin Castro, Texas Democrat, told NBC’s “Meet the Press” program Sunday. “I know there are a lot of folks who don’t believe that, people who don’t like to hear it. But the fact is the net migration with Mexico is at zero.”

Mr. Camarota countered that an equilibrium isn’t good enough. He said if the U.S. were enforcing its laws, the number of illegal immigrants would actually go down.

“Saying this huge problem is not a problem because the overall population isn’t getting bigger is like an obese person saying he doesn’t have a problem because he’s not getting any fatter,” Mr. Camarota said. “If we had just not let new illegals settle in the country, we would have reduced the population by 25 or 30 percent since 2009. That’s what these numbers imply.”

Mr. Obama has argued that the gains that can be made by enforcement alone are maxed out, and it’s time to legalize most illegal immigrants, both because the immigrants deserve it and because wiping the slate clean would allow authorities to go after the bad actors among the current population, and to focus on stopping future illegal immigrants.

A bill designed along those lines cleared the Senate in 2013, but Democratic leaders, who controlled that chamber at the time, never sent it to the House for action.

The president defended the legislation by saying the U.S. doesn’t have the will power to round up and deport most illegal immigrants, so they should be granted a place here.

Mr. Camarota said his research shows that there is another path: Stepping up enforcement and allowing natural attrition to reduce the numbers.

Part of the difficulty of the immigration debate lies in simple agreement on the facts.

Mr. Castro’s assertion Sunday that the border is secure was challenged by Rep. Raul R. Labrador, Idaho Republican, who said the San Francisco case is proof of a porous border.

“We do have an illegal immigration problem in the country,” he said.

He and Mr. Castro couldn’t even agree on the definition of “amnesty” for illegal immigrants. Mr. Labrador said it was granting illegal immigrants the thing they broke the law to achieve — a legal place in the U.S.

Mr. Castro denied that, describing amnesty as a no-questions-asked blanket forgiveness — something he said isn’t involved in the current talks.