The former Southern Cal running back would become the first player in the 75-year history of the award to have the Heisman Trophy taken away. The report also said the award would be left vacant for '05.
The NCAA found major violations in USC's football program and levied serious sanctions against the school in June.
"I can tell you the Heisman Trophy trust has made no decision regarding the Reggie Bush situation," Robert Whalen, executive director of the Heisman Trophy Trust, told The Associated Press on Tuesday.
The website cited two anonymous sources close to the Heisman Trophy Trust, who say the group's investigation is almost complete and would agree with the NCAA's finding that Bush was ineligible during the '05 season.
Typically, the Saints have declined comment on Bush's NCAA violations because they have no bearing on his pro career.
Team headquarters also were closed to reporters on Tuesday.
Bush's Los Angeles-based attorney, Shawn Chapman Holley, did not immediately respond to a request for comment from the AP.