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Bush, 88, a Republican who during his one term in office led a coalition that ejected Iraqi forces from Kuwait in 1991, was admitted to Methodist Hospital Nov. 23 for bronchitis.
He was transferred to intensive care on Sunday (local time) after setbacks including a persistent fever, family spokesman Jim McGrath has said.
"I don't have any guidance so far today except to say no news is good news," McGrath said on Thursday.
Hospital spokesman George Kovacik said the former president remained in intensive care on Thursday.
McGrath on Wednesday described Bush as alert and talking to medical staff.
Bush has lower-body parkinsonism, which causes a loss of balance, and has used wheelchairs for more than a year.
The 41st U.S. president, and father of former President George W. Bush, served as a congressman, as ambassador to the United Nations, as envoy to China, as CIA director and as vice president for two terms under Ronald Reagan.