The newspaper reports Giffords is mouthing the words to song lyrics, including "Twinkle, Twinkle, Little Star."
The congresswoman even spoke briefly on the phone with her brother-in-law aboard the International Space Station, according to the New York Times.
Giffords' chief of staff says the congresswoman definitely understands what is being said and has been offering short, verbal responses.
Giffords was shot in the head on January 8 while meeting with constituents in Tucson. Six people were killed in the attack, including a 9-year-old girl and a federal judge.
The Associated Press contributed to this story.