The former US president was admitted on Sunday after being taken into hospital on 23 November with bronchitis.
George H W Bush, the 41st president and father of George W Bush and Jeb Bush, who have both visited their father, has lower body parkinsonism, which causes a loss of balance. He has used a wheelchair for more than a year.
Jim McGrath, family spokesman, said: “The president is alert and conversing with medical staff, and is surrounded by family.
“Following a series of setbacks including a persistent fever, President Bush was admitted to the intensive care unit at Methodist Hospital on Sunday where he remains in guarded condition.
“Doctors continue to be cautiously optimistic about the current course of treatment.”
A hospital spokesman said in mid-December that the former president was expected to be home from in time for Christmas, but that spokesman later said doctors felt he should build up his energy before going home. He was joined on Christmas day by his wife, Barbara Bush.
The former president took office in 1989 and served one term in the White House. He also served as a congressman, ambassador to the United Nations, envoy to China, CIA director and vice president for two terms under Ronald Reagan.