The Arizona Daily Star reported the announcement was made by doctors at University Medical Center. The hospital said in a statement that the 40-year-old Arizona Democrat, who had been in critical condition the past week, was doing well and breathing on her own.
U.S. Sen. Kirsten Gillibrand, D-N.Y., said on NBC's "Meet the Press" Sunday she got an update on Giffords from Giffords' husband, Mark Kelly, the night before.
"She's using both sides of her body. She's able to breathe on her own. She's able to open her eyes and to show people she understands what she's hearing and seeing," Gillibrand said. "So … it's an extraordinary amount of progress for a woman who sustained such a horrific injury that she did."
Giffords, who was among 13 people wounded and six killed at an outdoor political gathering Jan. 8, was removed from a ventilator Saturday.
The newspaper said two victims remain at the hospital in good condition.
The presumed gunman, Jared Loughner, 22, of Tucson, remains in custody.
A private memorial service was held Sunday for Dorwan Stoddard, 76, one of the six who were killed, Tucson's WSAZ-TV reported. Stoddard was fatally wounded as he knocked his wife Mavy to the ground to save her.
The Arizona Daily Star reported the couple were seventh in line to see the congresswoman when the shooting started.
"She is heartbroken," Angela Robinson-Hellwege said of her mother, who has been released from the hospital. "But he died for her, so she wants to live for him."
In a related development, Tucson's KGUN-TV reported Pima County sheriff's spokesman Jason Ogan said Sunday James Eric Fuller, 63, one of the 13 wounded, was still being held at a Tucson hospital on an involuntary psychiatric hold for a mental evaluation.
Fuller was taken in Saturday afternoon by sheriff's deputies who arrested him on misdemeanor charges after he allegedly told Tucson Tea Party founder Trent Humphries during a town hall event: "You're dead."
Meanwhile, about 500 people took part in a walk for peace in Tucson Sunday morning, strolling about 2 miles from a park to the memorial outside Giffords' office, the Los Angeles Times reported.
Nancy Levy, 44, who brought her 9-year-old daughter Maya with her on the walk, told the Times she and a friend were participating "to make something positive out of something so negative, maybe make people think about being kind to each other."
Likewise, Blaze Mason, a 56-year-old freelance book editor, said the walk was a way for the community to make a statement.
"This way, we at least feel like we made some action that can help to heal the whole planet," she said.
Notable deaths of 2014 [PHOTOS]