"President Bush continues to improve as he recovers from pneumonia," spokesman Jim McGrath said in a statement.
"Despite a lingering cough, his lungs are clearing up and he is working with physical therapists to build strength. At his current rate of recovery, president Bush's physicians believe he could possibly go home over the weekend," he added.
Bush, 92, sought care at the intensive care unit of Houston Methodist Hospital on January 14 for "an acute respiratory problem stemming from pneumonia.
Former First Lady Barbara Bush, 91, was also admitted the same day to address a case of bronchitis after "experiencing fatigue and coughing," McGrath said.
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After a successful course of treatment involving antibiotics, the former first lady was discharged Monday.
Bush, who is the nation's oldest living ex-president, was invited to attend inauguration of his fellow Republican Donald Trump as president. He ended up watching the proceedings from his hospital room and sent Trump, whom he did not endorse as a candidate, a letter of support, McGrath said.
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