PHOENIX (Reuters) - A Mexican man pleaded guilty on Tuesday to first-degree murder in the 2010 death of a U.S. Border Patrol agent, a killing linked to a controversial U.S. attempt to track gunrunning to Mexico drug cartels.

Manuel Osorio-Arellanes made a deal with prosecutors and entered the plea in U.S. District Court in Tucson in the shooting death of agent Brian Terry in the Arizona borderlands, federal officials said. He could face life in prison, but prosecutors will not seek the death penalty.

Osorio-Arellanes, 36, was one of five Mexican suspects charged in Terry's murder. His death was linked to a U.S. Bureau of Alcohol, Tobacco, Firearms and Explosives investigation that allowed weapons to slip across the border into Mexico.

Two guns found at the remote area north of Nogales where Terry was killed in a shoot-out were traced to the sting operation, dubbed "Fast and Furious."

It remained unclear if the weapons were used in the murder and it is not known which of the five suspects killed the Border Patrol agent in the nighttime gunfight.

Laura Duffy, U.S. attorney for the southern district of California, said the deal represented "an important step in seeking justice on behalf of Agent Terry."

"Agent Terry was killed in the line of duty courageously safeguarding our border," Duffy, who is handling the case, said in a statement. "Our country owes him and his family a great debt of gratitude for his ultimate sacrifice in service to our country."

Osorio-Arellanes is scheduled to be sentenced on January 11. His attorney declined to comment on the plea agreement.

Arizona is a busy corridor for Mexican smugglers hauling marijuana and other drugs north to consumers in the United States.

According to the plea, Osorio-Arellanes admitted to entering Arizona in December 2010 with the others to rob traffickers of their drugs. Then on the evening of December 14, members of the group exchanged fire with agents in the area.

Osorio-Arellanes was wounded in the firefight and was arrested on the night of the shooting, federal prosecutors said.

Authorities offered a reward of up to $1 million in July for information leading to the arrest of the four other men believed to be involved in Terry's killing. Mexican police detained one of those men, Lionel Portillo-Meza, in September. He is awaiting extradition to the United States, prosecutors said.

The border killing sparked an election year firestorm between the administration of President Barack Obama and congressional Republicans, who slammed the government for allowing the program and called on Attorney General Eric Holder to resign.

The Republican-controlled House of Representatives found Holder, the nation's top law enforcement official, in contempt for withholding documents related to the gun-running probe.

Terry's family has filed a $25 million wrongful-death claim against the U.S. government, saying he was killed because federal investigators allowed guns to fall into the hands of violent criminals.

In a statement issued on Tuesday, the family said it was "extremely pleased" with the development, but noted that "three other ... fugitives are believed to be on the loose in Mexico and have thus far avoided capture" despite the bounty on their heads.

(Editing by Tim Gaynor, Cynthia Johnston and Mohammad Zargham)