Bush, 88, the nation's oldest living former president, was admitted to Methodist Hospital Nov. 23 including a week in intensive care last month.
"We are hopeful the President will be released early this week, but we are awaiting word from the doctors," spokesman Jim McGrath said.
Additional details would be provided "at the appropriate time," McGrath said.
Bush already had been in the hospital about a month for treatment of the persistent cough in December when his office disclosed he was in intensive care because physicians were having difficulty controlling a fever that developed after the cough was mostly resolved.
On Dec. 29, Bush's office said the former president had improved and was transferred back to a regular hospital room. Since then, his condition has continued to improve and he has been undergoing physical therapy to rebuild his strength.
Bush and his wife, Barbara, live in Houston during the winter and spend their summers at a home in Kennebunkport, Maine. On Jan. 6, they celebrated their 68th wedding anniversary. They are the longest-married presidential couple.
Bush had served two terms as Ronald Reagan's vice president when he was elected in 1988 to be the nation's 41st president. Four years later, after a term highlighted by the success of the 1991 Gulf War in Kuwait, he lost to Democrat Bill Clinton.
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