Homeland Security Secretary Janet Napolitano is under pressure to beef up border security in the Southwest in the wake of Saturday's killing of a rancher in southeastern Arizona.

Three members of New Mexico's congressional delegation have asked for an increase in the Border Patrol's presence in the Boot Heel of New Mexico, about 10 miles from where the rancher was shot to death over the weekend. U.S. Sens. Jeff Bingaman and Tom Udall, along with Rep. Harry Teague, say Napolitano's agency needs to take more security steps.

And former Colorado Rep. Tom Tancredo, an outspoken opponent of illegal immigration, called on Napolitano to "reject politics and do the right thing" by dispatching the National Guard to the Arizona border.

Cochise County Sheriff's Office deputies and detectives responded to an area northeast of Douglas on Saturday after searchers found the body of 58-year-old Robert Krentz inside his all terrain vehicle on his property. Detectives were able to determine that Krentz apparently came upon one person when he was fatally shot and his dog was wounded.

Cochise County investigators said Monday that Krentz likely was killed by an illegal immigrant, but there's no evidence to suggest there was any confrontation that led to the shooting.

Tancredo, who attended a Tea Party event over the weekend in Arizona, blasted Napolitano for not doing more to secure the border.

“As governor of Arizona, Napolitano deployed the National Guard to help the Border Patrol do its job… Three days ago, Napolitano told an audience at Arizona State University that the border is more secure than ever," Tancredo said Sunday through his Rocky Mountain Foundation. "I challenge her -- no, I dare her -- to come to this community and try to sell that lie.”

U.S. Rep. Gabrielle Giffords, D-Ariz., whose Congressional district covers the area, described the rancher as a pillar of the community who was recently inducted into the Arizona Farming and Ranching Hall of Fame.

"The cold-blooded killing of an Arizona rancher is a sad and sobering reminder of the threats to public safety that exist in our border communities," Giffords said. "It has not yet been determined who committed this atrocity or why, but I know that federal and local authorities are mobilizing every possible resource to locate and apprehend the assailant."

Giffords said if Krentz's killing is connected to drug cartels or smugglers, the federal government must respond appropriately.

"All options should be on the table, including sending more Border Patrol agents to the area and deploying the National Guard," Giffords said.

Arizona Attorney General Terry Goddard said government has a clear responsibility to aid law enforcement resources at all levels along the border.

"I call on our federal and state governments to work together to bolster the law enforcement resources needed to better protect Arizonans living on the border," Goddard said

U.S. Border Patrol spokesman Omar Candelaria told the The Arizona Daily Star that if Krentz's killing was tied to such border crime, it would be a first for the area, to his recollection.

At a news conference Monday, Cochise County Sheriff Larry Dever said Krentz was out checking water line and fencing on the land Krentz's family has ranched since 1907. Dever says Krentz had weapons with him in his all-terrain vehicle but didn't use them.

Investigators say Krentz apparently came upon one person when he was shot. While Krentz was still in his vehicle, mortally wounded, he managed to drive the ATV away from the scene at a high rate of speed before becoming unconscious.

Sheriff's deputies, U.S. Border Patrol trackers and Department of Corrections dog chase teams followed footsteps approximately 20 miles south to the Mexican border. No suspects have been apprehended.

The Associated Press contributed to this report.