A man suspected of shooting the Idaho pastor who led a prayer at a rally for US Republican presidential candidate Ted Cruz was arrested on Tuesday and was facing attempted murder charges, police said.

Kyle Odom, 30, was taken into custody in Washington DC in connection with the Sunday afternoon shooting of Pastor Tim Remington outside the Altar Church, Coeur d'Alene, Police Department Chief Lee White said.

Odom was apprehended by Secret Service officers after he tossed several items, including flash drives, over the fence of the White House, White said.

Local broadcaster KXLY reported that a Facebook page linked with Odom, who police said suffered from mental illness, was updated on Tuesday with a statement claiming that Remington was part of an ancient Martian civilization that ruled Earth.

Remington had led a prayer at a Cruz rally on Saturday and was shot by Odom, an ex-marine, the next day in the church parking lot in a preplanned attack, police said.

The senior pastor was shot six times, including in the skull, after the Sunday morning service, John Padula, outreach pastor at the church told Reuters. Remington regained consciousness on Monday evening as a candlelight vigil for his recovery was underway, Padula said.

"He opened one eye and gave me a thumbs up," Padula said, adding that Remington does not have feeling in his right arm but appears to be improving. "Without God, there is no way he'd be here."

Cruz, an outspoken supporter of gun rights, could not be reached for comment on Tuesday, but a campaign spokeswoman told NBC News on Monday that they were praying for Remington's recovery.

"Our prayers are with Pastor Tim, his family, and the doctors who are supervising his care," Catherine Frazier said. "We pray for his full recovery and are thankful for the efforts of law enforcement to ensure the attacker is swiftly brought to justice."

One church member said: "We just want to be here for everyone in our congregation. Be here for our pastor. We are going to go to the hospital and wait for the word to make sure he is okay."

"He opens his home to people. He takes them in and lets them live in his home," another church member said. "All the people society has turned their backs on, he goes out and he reaches out to them."

White read a statement from Odom's family saying they were thankful for his "safe apprehension".

Additional reporting by Reuters.