"The President is alert and, as always, in good spirits - and his exchanges with doctors and nurses now include singing," family spokesman Jim McGrath said in a brief statement.
The 88-year-old Bush, the nation's oldest living former president, was admitted at Methodist Hospital in Houston on Nov. 23 because of a bronchitis-related cough, after spending about a week there earlier in November for the same condition.
The cough was mostly resolved by the time he was moved to intensive care on Sunday for treatment of a fever that doctors were having difficulty controlling.
"The Bushes thank everyone for their prayers and good wishes and, like their doctors, are cautiously optimistic that the current course of treatment will be effective," McGrath said.
On Thursday, a longtime Bush aide tried to quell concern about Bush's condition by saying the former president likely would advise well-wishers to "put the harps back in the closet."
Jean Becker, Bush's Houston chief of staff, said the former president would likely be in the hospital for a while, noting his age and that "he had a terrible case of bronchitis which then triggered a series of complications."
Becker said "most of the civilized world" contacted her after word spread that Bush had been placed in intensive care unit.
"Someday President George H.W. Bush might realize how beloved he is, but of course one of the reasons why he is so beloved is because he has no idea," Becker said in the at-times lighthearted statement that made multiple references to jokes and the former president's sense of humor.
Updates on his health are being limited out of respect for the family, she said.
The family had hoped Bush would have been well enough to spend Christmas at home. His cough eased, but he developed a persistent fever. His condition has since been downgraded to "guarded."
The former president has been visited by family and friends, including longtime friend James Baker III, his former Secretary of State. Bush's daughter, Dorothy, arrived Wednesday from her home in Bethesda, Md. Other visitors have included his sons George W. Bush, the 43rd president, and Jeb Bush, the former Florida governor.
Bush and his wife, Barbara, live in Houston during the winter and spend their summers at a home in Kennebunkport, Maine.
Bush, the 41st president, had served two terms as Ronald Reagan's vice president when he was elected in 1988. Four years later, after a term highlighted by the success of the 1991 Gulf War in Kuwait, he lost to Democrat Bill Clinton amid voters' concerns about the economy.
Bush was a naval aviator in World War II - at one point the youngest in the Navy - and was shot down over the Pacific. He's skydived on at least three of his birthdays since leaving the White House, most recently when he turned 85.
Bush, also a former U.S. ambassador to China and CIA director, suffers from a form of Parkinson's disease that forced him in recent years to use a motorized scooter or wheelchair for mobility.