Methodist Hospital spokesman George Kovacik said Saturday there is no change in the status of the 88-year-old Bush from earlier in the week. But Jim McGrath, a spokesman for the 41st president, said "he still has the cough and we're now not expecting him to be discharged this weekend."
McGrath said doctors are "going to play it safe there for obvious reasons."
Aides had expected his discharge soon, but McGrath said "the cough has taken longer to clear up" than anticipated.
Bush has been in and out of the hospital recently for complications resulting from bronchitis. He has been treated with antibiotics and steroids, and this marks Bush's second extended hospital stay since November.
Hospital visitors have included Bush's son, former President George W. Bush, and his wife Laura, who live in Dallas.
The former president has a form of Parkinson's disease that has forced him to use a wheelchair or motorized scooter to get around. Bush was seen a few weeks ago attending a Houston Texans NFL game, something he does frequently.
The elder Bush was vice president under Ronald Reagan from 1980 to 1988, when he was elected president. He lost his re-election bid to Bill Clinton four years later.
Bush and his wife, Barbara, now make their winter home in Houston and spend summers in Kennebunkport, Maine.
The former president was a naval aviator in World War II -- at one point the youngest in the Navy -- and was shot down over the Pacific. He achieved notoriety in retirement for skydiving on at least three of his birthdays since leaving the White House in 1992.