Four women have come forward to accuse George Bush of touching them inappropriately, including one who said this week that the 93-year-old former president grabbed her years before he began using a wheelchair.

After the second allegation, Mr. Bush’s office on Wednesday night released a statement that both acknowledged the stories that had emerged and offered an apology.

“At age 93, President Bush has been confined to a wheelchair for roughly five years, so his arm falls on the lower waist of people with whom he takes pictures,” the statement said.

“To try to put people at ease, the president routinely tells the same joke — and on occasion, he has patted women’s rears in what he intended to be a good-natured manner. Some have seen it as innocent; others clearly view it as inappropriate. To anyone he has offended, President Bush apologizes most sincerely.”