"As much as it frustrates me, this process takes time," McConnell said in a statement on Sunday, his first since he was hospitalised last month.
"And on the advice of my doctors, I won't be able to return to the Senate floor to vote quite yet."
He said that he suffered a fall in mid-June that left him briefly unconscious.
While hospitalised, he developed pneumonia and was treated with antibiotics, according to a separate statement that McConnell's office attributed to the attending physician.
The physician was not identified.
McConnell, 84, said he is now at a rehabilitation centre, which he did not name.
He will focus on "physical therapy and strategies to reduce his risk of future falls", according to the physician's statement.
The Kentucky Republican and former Senate majority leader, who now chairs the Senate Rules Committee, said he has been working with his legislative staff on current issues and keeping in touch with Senate colleagues.
McConnell has been out of public view since mid-June, when he was taken from his home to a hospital in the Washington area for reasons that were not disclosed until the latest statement.
Less than a day earlier, the office of US Senator Lindsey Graham, a South Carolina Republican and one of the chamber's most prominent members, announced that he had died from a heart ailment.
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