The 88-year-old Bush has reportedly been suffering from persistent respiratory issues and was checked in to the hospital on the day after Thanksgiving.
"His big problem is a chronic cough, he can't get rid of so he's back at Methodist," his chief of staff, Jean Becker told the Chronicle. "This is not a life-threatening illness."
Bush spokesman Jim McGrath told CNN that his family hopes he will be discharged by the weekend.
The Associated Press reports that Bush's son, former President George W. Bush, has been among visitors.
More from AP:
"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."
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.
This story is developing. Check back for more updates...