President Trump’s lawyers do not know just how much the White House counsel, Donald F. McGahn II, told the special counsel’s investigators during months of interviews, a lapse that has contributed to a growing recognition that an early strategy of full cooperation with the inquiry was a potentially damaging mistake.

The president’s lawyers said on Sunday that they were confident that Mr. McGahn had said nothing injurious to the president during the 30 hours of interviews. But Mr. McGahn’s lawyer has offered only a limited accounting of what Mr. McGahn told the investigators, according to two people close to the president.

That has prompted concern among Mr. Trump’s advisers that Mr. McGahn’s statements could help serve as a key component for a damning report by the special counsel, Robert S. Mueller III, which the Justice Department could send to Congress, according to two people familiar with the discussions.

Mr. Trump’s lawyers realized on Saturday that they had not been provided a full accounting after The New York Times published an article describing Mr. McGahn’s extensive cooperation with Mr. Mueller’s office. After Mr. McGahn was initially interviewed by the special counsel’s office in November, Mr. Trump’s lawyers never asked for a complete description of what Mr. McGahn had said, according to a person close to the president.