Bush, 88, has been in and out of the hospital recently for complications resulting from bronchitis, Methodist Hospital said in a brief statement.
The hospital and Bush's spokesman in Houston, Jim McGrath, described Bush as in stable condition and said they expected the former president to be released by the weekend. He's been under hospital care for nearly a week.
"If you asked him today, he would tell you he feels good enough to get out this afternoon," McGrath said Thursday. "But the doctors have a different view. He's 88, and they're being extra careful, and understandably so."
The former president's illness was described as not life-threatening.
"Any time someone the president's age had bronchitis, there's concern about possible pneumonia," Dr. Amy Myunderse, who is in charge of Bush's care, told the Houston Chronicle. "Mr. Bush's condition never progressed to that level."
Bush was treated with antibiotics and steroids, she said.
McGrath said this is Bush's second extended hospital stay this month.
"This cough has kind of been building on itself since roughly the beginning of the month," he said. "He was hospitalized earlier in the month for a little over a week. They thought they had successfully contained the bronchial issues but they hadn't.
"It kind of flared up again over Thanksgiving and the day after Thanksgiving, he was readmitted."
McGrath said physicians want the cough to go away completely before Bush is discharged."They don't want to put him through this again," McGrath said.
Bush's son, former President George W. Bush, and his wife, former first lady Laura Bush, slipped in to see him Sunday. They now live in Dallas.
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.
Bush was seen a few weeks ago attending a Houston Texans NFL game, something he does frequently. He suffers a form of Parkinson's disease that has forced him to use a wheelchair or motorized scooter to get around.
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.