Law enforcement agencies were pursuing "a number of leads" in the United States and in Mexico but no one had been arrested or charged with killing Agent Robert Rosas, the FBI said late Friday.
Investigators said they have notified hospitals on both sides of the border to be alert for patients with suspicious or unexplained injuries.
The Los Angeles Times reported that police in Tecate, Mexico, said Friday they had arrested an injured man walking near the crime scene with a Border Patrol-issued weapon shortly after the shooting. The man, Ernesto Parra Valenzuela, 36, was taken to a hospital, according to a news release.
After-hours messages left for the FBI were not immediately returned.
Rosas, 30, was killed Thursday night while responding alone to a suspected border incursion near Campo, a town in rugged, arid terrain in southeastern San Diego County. He was shot in the head and body and was dead when backup agents arrived, said Keith Slotter, special agent in charge of the FBI's San Diego bureau.
Federal officials have expressed concerns that the drug cartel battles plaguing Mexico could spill into the United States with the targeting of U.S. law enforcement officials. Slotter said investigators aren't ruling out the possibility that Rosas was slain by drug smugglers or even human smugglers.
Investigators said blood evidence at the scene indicated at least one suspect and possibly more had serious injuries, perhaps by gunfire.
Investigators don't yet know how many shots were fired, if Rosas fired any shots himself and how many guns were used.
"It's all possible. I can't definitively say X number of people fired or Agent Rosas got off shots or didn't. I mean, it's too early in the investigation to say that with any certainty," Slotter said.
Authorities said at least one other agent in the field heard gunshots after Rosas left to respond to the call and couldn't reach Rosas on his radio afterward.
Rosas was the first Border Patrol agent to die in a shooting in more than a decade, according to the Officer Down Memorial Page Inc., which tracks fallen officers using information provided by law enforcement agencies. Another agent, Luis Aguilar, was intentionally run over by a fleeing man driving a drug-laden Hummer in January 2008.
Rosas, a three-year Border Patrol veteran, had a 2-year-old son and an 11-month-old daughter, said Richard Barlow, acting chief patrol agent for the Border Patrol's San Diego sector.
"My thoughts and condolences are with Agent Rosas' family and his fellow agents at this difficult time," said Department of Homeland Security Secretary Janet Napolitano said in a statement. "His death is a vivid reminder that we are engaged in a serious effort to secure our border and that thousands of Border Patrol agents and other DHS employees risk their lives every single day to protect and defend our nation."
Barlow said he could not confirm reports that Rosas called for backup and then went ahead before anyone arrived. But he said it isn't unusual for agents to work alone along the 60 miles of border in the San Diego sector.
"It is a common occurrence for our agents to start tracking individuals or start pursuing individuals that make an incursion into the United State by himself prior to backup arriving," he said.
The San Diego sector of the Border Patrol has seen a 22 percent decrease in border apprehensions this year after a 7 percent increase in each of the previous two years. Barlow said agents routinely have rocks thrown at them and are physically assaulted.
The president of the union representing 17,000 Border Patrol agents declined to discuss the details of the shooting but said his organization has long been concerned about staffing levels and situations where agents work alone in the field.
T.J. Bonner, president of the National Border Patrol Council, said it was "fairly common for our agents throughout San Diego County and the rest of the country to work without a partner. They each have separate vehicles, and it's a matter of concern with us."
Since 1919, 108 Border Patrol agents have died on duty, according to The Officer Down Memorial Page Inc. Gunfire was the leading cause with 30 deaths, followed by automobile accidents and aircraft accidents.
The FBI is offering a $100,000 reward for information that leads to the arrest and conviction of a suspect or suspects.
Associated Press Writers Christina Hoag in Los Angeles and Amy Taxin in Long Beach contributed to this report.