The city of Las Cruces, the second largest in the US state of New Mexico, remains in a state of unease as the FBI and other law enforcers remain clueless on the identities of the culprits who planted bombs in three churches in the city and their possible motive.

Two of the bombs exploded this month during Sunday services at the Holy Cross Roman Catholic Church and Calvary Baptist Church while an undetonated bomb was discovered last Friday at a Presbyterian church, according to the New York Times.

A surveillance video at the Calvary Baptist Church showed a young man walking at the parking lot before the bomb exploded at about 8:20 a.m. last Aug. 2.

The FBI is offering $20,000 reward for information that will lead to the resolution of the case.

"We're still connecting the dots," said Las Cruces police chief Jaime Montoya.

Authorities are investigating all angles and are questioning political activists, members and neighbours of the churches and witnesses.

The explosions have authorities on high alert with bomb-sniffing dogs used last Sunday on the three churches.

Last Tuesday, bomb threats were made at the Last Cruces District Courthouse and the city's adult probation office, which were found to be negative.

Las Cruces has a mix of Hispanics and Caucasians and is near the Mexican border.

"We're not split by railroad tracks, with one group on one side and another group on the other side," Montoya said.

The city has become a battleground for abortion rights following the establishment of an abortion clinic, which is near El Paso, Texas, which has tough anti-abortion laws.

Lucas Herndon, who works for pro-abortion rights group ProgressNOW New Mexico, was asked by the FBI to hand over a video he had recorded during a recent anti-abortion rally.

"The agents were interested in extremists from both sides," he said. "But they didn't seem to have a very clear idea of who did it or why."

FBI assistant special agent Robert White told parishioners at the Holy Cross Roman Catholic Church last Sunday to "look back one day, look back one week, for anything out of the ordinary" before the explosion.

"It's our intent to fully identify the individual or individuals, if there is more than one, and bring him to justice," White said.

Stephan Marshall, the chief division counsel for the FBI in Albuquerque, said, "at this point, we haven't gotten anything definitive."