The Brumbies were not immediately available to confirm the reports, though chief executive Andrew Fagan told the Australian Broadcasting Corporation they were disappointed to lose the services of the South African.
The 2007 World Cup winning coach had made no secret of his desire to move back into the international coaching arena, having missed out on the vacant Wallabies position in early July.
"We understand how bitterly disappointed he was on missing out on the Wallabies coaching job," Fagan told the ABC. "That's driven his decision.
"He felt strongly that (getting another international coaching job) was no longer going to be possible in Australia and at the same time he felt the club was in a position far stronger than he thought it would be after two years.
"He thought this was as good as time as any to move on."
White's decision comes less than two months after he led the Brumbies to the Super Rugby final against the Waikato Chiefs, their first appearance in the title-decider since 2004.
Assistant coaches Laurie Fisher, who had been head coach previously, and former Wallabies flyhalf Stephen Larkham are reported to be favourites to replace him.