A public announcement was expected later in the day.
There was no immediate word on who would be nominated to replace Moseley and Wynne.
Press secretary Dana Perino said President Bush knew about the resignations but that the White House "has not played any role" in the shake-up.
Moseley became Air Force chief in September 2005; Wynne took office in November 2005.
Wynne is the second civilian chief of a military service to be forced out by Gates. In March 2007 the defense secretary pushed out Francis Harvey, the Army secretary, because Gates was dissatisfied with Harvey's handling of revelations of inadequate housing conditions and bureaucratic delays for troops recovering from war wounds at Walter Reed Army Medical Center.
The Air Force has endured a number of embarrassing setbacks over the past year. In August, for instance, a B-52 bomber was mistakenly armed with six nuclear-tipped cruise missiles and flown across the country. The pilot and crew were unaware they had nuclear arms aboard.
The error was considered so grave that President Bush was quickly informed.
Moseley later announced that in response to flaws exposed during the nuclear weapons error, the Air Force would change the way bomber crews organize for their nuclear training mission.
Gates also has been trying to learn more about how fuses for Air Force ballistic missiles were mistakenly shipped to Taiwan. Gates has been briefed on the conclusions of an internal investigation of that matter but the results have not been made public.