Doctors at Tucson's University Medical Center said the 40-year-old Arizona congresswoman is blinking. Both her eyes have been unbandaged following a medical procedure during the weekend, the Arizona Daily Star reported.
The newspaper said doctors Saturday relieved pressure and bulging that was forming in her right eye. Dr. Lynn Polonski said while the bullet went through the left side of her brain, the impact sent "shock waves through the brain" and the right side of her skull also was damaged.
Polonski said the fact Giffords can focus both eyes on her husband, astronaut Mark Kelly, and track his movement is a good sign.
Giffords' breathing tube has been removed from her face and replaced with one that goes through her throat. Her condition was upgraded from critical to serious Sunday. She has some movement on her right side, though doctors have not been specific about that part of her recovery.
While it's not yet known when Giffords will be moved to rehabilitation, Kelly said the family is looking at facilities across the country,
"The day she leaves this hospital, that's her graduation," The Arizona Republic quoted Dr. Michael Lemole as saying.
The case of Jared Loughner, the 22-year-old Tucson man charged with killing six people and wounding 12 others, including Giffords, during a public gathering outside a Tucson grocery store Jan. 8, will be moved to San Diego from Tucson due to extensive pretrial publicity, The Washington Post reports.
Chief Judge Roslyn O. Silver has not officially made the venue-change decision, but "it's going to happen. It's just a matter of time," a law enforcement source told the Post.
San Diego will get the case, in part, because it's one of the closest judicial districts to Arizona, the Post said.
U.S. District Judge Larry Alan Burns of San Diego was appointed last week to hear the case because Arizona judges recused themselves. Judy Clarke, Loughner's attorney, is also based in San Diego.
Venue changes are not regularly granted, the Post said, but they have occurred in high-profile cases.
In 1996, a federal judge moved the Oklahoma City bombing case to Denver, saying defendants Timothy J. McVeigh and Terry L. Nichols had been "demonized" in the media.
The April 19, 1995, bomb attack on the Alfred P. Murrah Federal Building in Oklahoma City -- which killed 168 people, including 19 children under age 6 -- was the most destructive act of terrorism on U.S. soil until the Sept. 11, 2001, attacks.
Loughner, charged with murder and attempted murder, is being held at the Federal Correctional Institution, a medium-security prison outside Phoenix.