The 88-year-old former president's chief of staff, Jean Becker, said the condition was not considered life-threatening, the Houston Chronicle reported.
Concerns that the cough would develop into pneumonia have eased and Bush was expected to be discharged this weekend, she said.
"President Bush is in the hospital," Becker said. "We have kept this quiet out of respect for him."
Becker said Bush had made repeated visits to Methodist Hospital in recent weeks, and was most recently admitted on the day after Thanksgiving, the newspaper said.
"His big problem is a chronic cough he can't get rid of so he's back at Methodist," Becker said. "This is not a life-threatening illness."
Former President George W. Bush and his wife, Laura, have been among the elder Bush's visitors.
George H.W. Bush was vice president for two terms under President Ronald Reagan before serving as president from 1989-1993.