Alaniz said Ortiz is suspected of shooting the four victims, two of whom were identified Saturday as Melissa Ramirez, 29, and Claudine Luera, 42.
Officer Alaniz said the dead are believed to have been prostitutes and that one of them was a transgender woman.
"The arrest of the border patrol agent who has allegedly committed multiple murders could not have happened without the hard work and expertise of the Texas Rangers, the Webb County Sheriff's Office, the Webb County District Attorney's Office, and Laredo Police Department".
Alaniz described Ortiz as a serial killer.
The Texas Rangers, along with the Webb County Sheriff's office, launched an investigation after Melissa Ramirez, 29, a mother of two, was found slain September 4.
Boston's Mookie Betts leaves game after getting hurt
For more on Sale, check out the "Red Sox Final" video above, presented by Rodenhiser Heating & Cooling. Betts walked off slowly and was replaced by Tzu-Wei Lin, who went into centre.
The spree came to an end Friday when a fifth woman Ortiz pulled his gun on fought him off, hopped out of his vehicle and ran to a gas station pleading for help, according to the report.
She pulled off her shirt and ran to a nearby gas station, where she found a state trooper.
He has admitted the slayings to cops, the paper reported, citing a police affidavit. According to a police affidavit, Ortiz said he killed Ramirez, a mother of two, the previous day.
U.S. Customs and Border Protection officials continue to refuse to release information about how many Border Patrol agents have been assaulted this fiscal year. They believe Ortiz acted alone.
Authorities found Ortiz at a gas station in Laredo, about 160 miles southwest of San Antonio. CBP says in a statement that while it is the agency's policy not to comment on an ongoing investigation, "criminal activity by our employees is not, and will not be tolerated".
Quelles sont les émissions en ligne les plus demandées par les utilisateurs?
Les disciplines sportives les plus populaires Les émissions sportives en direct sont de plus en plus accessibles. Récemment, les émissions sportives sont de plus en plus populaires sur Internet.
"Our sincerest condolences go out to the victims' family and friends", Meehan said.
Ortiz is being held on $2.56 million bond. He was quoted by the Laredo Times newspaper.
Alaniz is also overseeing another case involving a Border Patrol agent in Laredo, a man accused of killing his lover and their 1-year-old son.
Investigators discovered two more bodies, one on Friday and one on Saturday in a remote area along Interstate Route 35.
Mr Alaniz said the cause of death was similar in each case.
First Hurricane Florence deaths: mother, infant killed by tree
Henry McMaster has not yet lifted the evacuation order he put in place Tuesday, and it is not clear when the order will be lifted. In Hampstead, North Carolina, emergency responders going to a call for cardiac arrest found their path blocked by downed trees.
Detectives asked him to do "the right thing", Alaniz said.