Larkham, who formed one of rugby's most successful halves partnerships with fellow Brumbies' player George Gregan, has been the team's attack coach for the past three seasons, helping the Canberra-based side to the final of this year's Super Rugby competition.
"You're never really sure if you're ready but my rugby knowledge is pretty good and I played a lot of rugby under some really good coaches," Larkham said in comments published by The Australian newspaper on Friday.
"It would have been nice to work under Jake for another year or so but things have changed and I'm really excited by the prospect."
South African White, who coached the Springboks to their 2007 World Cup triumph, walked out on the Brumbies this week with two years left on his contract.
In just two seasons, White transformed a cellar-dwelling team plagued by in-fighting into one of the most disciplined sides in the southern hemisphere regional competition, and his "pyschological" work with the young Brumbies roster would be missed, Larkham said.
"But the team has now had two years under this program and at some stage you have to assume responsibility and accountability," Larkham, 39, added.
The Brumbies will look at external candidates but are likely to opt for either Larkham or 55-year-old forwards coach Fisher, who has also expressed interest in the head role, with only a few months to bed down a new coach before the 2014 season.
Brumbies CEO Andrew Fagan said he was "shocked" by White's decision, which he put down to his disappointment at missing out on the Wallabies' head coaching role to incumbent Ewen McKenzie.
White has returned to South Africa, where local media have linked him to Super Rugby sides, the Durban-based Sharks and Cape Town's Stormers.
Clyde Rathbone, a South Africa-born former Wallabies back who linked up with former mentor White at the Brumbies, described the departed coach as a "highly strung, emotional and complex individual" who was "always chasing the next opportunity" in a blog post on Australian sports website The Roar (www.theroar.com.au).
"I know how disappointed Jake was at missing out on the Wallabies job. I know he felt as though the politics that marred his time with the Springboks and which he believed he had escaped in Australia had burnt him once more.
"With that said, failure to secure the Wallabies job is not an acceptable reason to resign from the Brumbies. It's not even close to being an acceptable reason."