Jim McGrath of The Methodist Hospital said Bush, 88, has been treated since Nov. 23 because of the cough, the Houston Chronicle reported Sunday.
"Spirits are high and he got a good night's sleep," McGrath said.
"But he's still got a little bit of a cough and until we get that under grips we're not looking at any discharge date."
The former president's doctors have said Bush was hospitalized as a precautionary measure and that his condition was never considered life-threatening.
Susan Sarandon 'very excited' about daughter's pregnancy
Scarlett Johansson steps out with fiance after pregnancy reveal