Investigators believe the handwritten message was a reference to Giffords, Pima County Chief Rick Kastigar said. The note was found in a safe alongside other ones, including "I planned ahead," "My assassination" and the name "Giffords."
Authorities revealed other new information on Tuesday about the events leading up to the Saturday shooting that killed six people and injured 14 others, including the three-term Democrat.
On Saturday morning, Jared Loughner's father saw him take a black bag out of a car trunk, Sheriff Clarence Dupnik told the AP.
The father approached Loughner, and he mumbled something and took off running, Dupnik said. The father got in his truck and chased his son as he fled on foot.
Loughner took a taxi cab to the supermarket where the three-term Democrat was holding a meeting to hear the concerns of her constituents, authorities said earlier. Among those killed were a federal judge and a 9-year-old girl.
Tuesday afternoon, Loughner's parents released a statement saying that they don't understand why the shooting happened and that they can't express their feelings in words.
"This is a very difficult time for us," read the statement from Randy and Amy Loughner, parents of suspected shooter Jared Loughner, 22, that was handed out to reporters standing in a shady driveway outside their Tucson home. "We ask the media to respect our privacy. There are no words that can possibly express how we feel. We wish that there were, so we could make you feel better. We don't understand why this happened."
Both parents have been interviewed extensively by the FBI as part of the investigation into Saturday's shooting in which six people, including a federal judge, died and 14 others, including U.S. Rep. Gabrielle Giffords, were wounded.
Earlier, the parents told federal investigators they knew their son was becoming increasingly troubled but were not aware how much he had drifted and were "completely surprised" that he actually committed a violent act, a law enforcement source told CBS News investigative producer Pat Milton Tuesday.
The source told Milton that one of the critical questions investigators initially tried to determine is whether Loughner was spurred on by hate groups or pushed or directed by some other individuals or groups. So far there has been no information that is the case.
The source also told Milton that investigators are looking into whether Loughner may have conducted surveillance on Giffords' district office and campaign rallies.
Loughner's parents are devastated, according to neighbor Wayne Smith, 70.
Smith shocked the Loughners with the news of their son's arrest this weekend as they returned from shopping. He told Tracy the parents can't stop crying.
"They're devastated," said Smith. "How would you feel if your son did it? You'd feel like that you're to blame for the people getting shot. They are just deeply grieved about it."