Immigration officials revealed on Monday that a fugitive Mexican national accused of killing five neighbors over the weekend had previously been deported four times. Even as he remained at large and the target of an extensive manhunt, the case seemed sure to reignite the bitter national debates over immigration policy and gun control.

It began Friday evening with a type of noise complaint not uncommon in rural Texas. The authorities said that the suspect, Francisco Oropesa, was shooting a gun in his yard in Cleveland, Texas, when a neighbor, Wilson Garcia, approached him and asked him to stop so that his baby could sleep.

Mr. Oropesa, 38, responded by getting an AR-15 rifle from his house and walking over to Mr. Garcia’s home about 11:30 p.m., where he killed Mr. Garcia’s 8-year-old son, wife and three other people, the authorities said.

Two women who were killed were shielding a 6-week-old boy and a 3-year-old girl. The gunman then chased Mr. Garcia, who escaped through a window and ran.

An official with United States Immigration and Customs Enforcement said on Monday that an immigration judge had ordered I.C.E. to deport Mr. Oropesa to Mexico in March 2009. Mr. Oropesa illegally returned to the United States, and he was caught and removed several more times by I.C.E. in September 2009, January 2012 and July 2016, the official said.

It was unclear what had led to his initial deportation order, but the immigration official said that Mr. Oropesa was later convicted in Montgomery County, Texas, for driving while intoxicated in January 2012 and sentenced to jail.

Even while the F.B.I. and several Texas law enforcement agencies sought the fugitive, attention turned quickly to the immigration status of the suspect and his victims.

Gov. Greg Abbott of Texas, a frequent critic of federal policy, said on Sunday that the suspect was in the country illegally but on Monday walked back part of his initial comments that the victims were “five illegal immigrants.”

“We’ve since learned that at least one of the victims may have been in the United States legally,” Renae Eze, a spokesman for the governor, said in a statement on Monday. “We regret if the information was incorrect and detracted from the important goal of finding and arresting the criminal.”

I.C.E. officials did not immediately respond to an email about the immigration status of the victims who were killed, all of whom were from Honduras.

Before the governor walked back part of his comments, the League of United Latin American Citizens, a civil rights organization, called for him to apologize for his remarks about the victims.

“L.U.L.A.C. firmly believes that every human being, regardless of their immigration status, deserves to be treated with dignity and respect,” said Rodolfo Rosales Jr., the Texas L.U.L.A.C. state director.

Gun-control advocates, meanwhile, were quick to point out that the suspect had used an AR-15 rifle, the weapon of choice in many shootings, which critics call a weapon of war designed kill as many as people as possible.

It was not known where the suspect had obtained the gun, which the authorities said had been discarded and found after the killings. It was possible, they said, that he still had a second weapon.

The authorities have offered a total reward of $80,000 for information leading to the capture of Mr. Oropesa, and on Monday posted signs in Spanish in Cleveland and elsewhere in San Jacinto County seeking help in finding him.

The search on Monday briefly extended outside Cleveland, Texas, which is about 45 miles northeast of Houston, when the authorities in neighboring Montgomery County received a report that a man who possibly matched a description of Mr. Oropesa had been seen in the area.

The Montgomery County Sheriff’s Office asked residents to stay indoors while deputies, dogs and helicopters scoured the area, but officials did not find him.

On Sunday, law enforcement officials conceded that they did not know the suspect’s whereabouts, adding that they considered him to be a threat.

“We do not know where he is,” James Smith, the special agent in charge of the F.B.I. in the Houston area, told reporters at a news conference. “We do not have any tips right now as to where he may be. Right now, we have zero leads.”

Adding difficulty to the search, the authorities had initially identified the suspect as Francisco Oropeza, but on Sunday afternoon the F.B.I. said that going forward his surname would be spelled as Oropesa “to better reflect his identity in law enforcement systems.”

An “incorrect” image of Mr. Oropesa had been “mistakenly disseminated,” the agency said on Twitter on Sunday. The F.B.I. said it had since removed the image from its social media accounts and asked that others not share it.

Sheriff Greg Capers of San Jacinto County said that there were 10 people inside the house at the time of the shooting. He said that Mr. Oropesa had been drinking when Mr. Garcia asked him to stop firing his gun. Sheriff Capers said that Mr. Oropesa responded, “I’ll do what I want to in my front yard.”

Sheriff Capers said the authorities believed that they had recovered the AR-15 used in the shooting, but that Mr. Oropesa could still be carrying another weapon. The authorities said they had found additional guns in his home as well as a phone.

The Federal Bureau of Investigation identified those killed as Mr. Garcia’s wife, Sonia Guzman, 25; Diana Velazquez Alvarado, 21; Juliza Molina Rivera, 31; Jose Jonathan Casarez, 18; and Daniel Enrique Laso, 8.

“I have no words to describe what happened,” Mr. Garcia said in Spanish at a vigil on Sunday evening, where dozens of people surrounded him and the other survivors of the shooting, joining them in prayer. “We are alive but there is no life,” he said. “I was able to escape by a miracle.”

Maria Jimenez Moya, Livia Albeck-Ripka, Eduardo Medina, Eliza Fawcett, April Rubin and Remy Tumin contributed reporting. Kirsten Noyes and Jack Begg contributed research.