The BBC's Chris Hogg, in Tokyo, says the resignations have led critics to complain that the party is out of touch and lacks a proper mandate to govern.
The LDP's recent slump in popularity has led some to suggest that the party could be on the verge of losing power - an almost unthinkable prospect for most of the past 50 years.
Our correspondent says there is little doubt that the LDP will select Mr Aso as its leader.
Newspaper polls have suggested that Mr Aso has secured two thirds of the votes from the party chapters and has the support of 60% of law makers.
Mr Aso, a former foreign minister, pledged to a crowd of supporters in Tokyo that he would sort out Japan's economy.
"America is facing a financial crisis... we must not allow that to bring us down as well," he said.
Bad old days?
The 68-year-old veteran is promising greater public spending to try to stimulate the economy - particularly in rural areas, where the party is traditionally strong.
But one of his rivals for the leadership, Kaoru Yosano, accused him of risking Japan's long-term interests through wasteful spending.
Other colleagues fear higher spending could mark a return to the old profligate ways of the LDP, where expensive public works projects were used to try to create jobs, hollowing out the public finances.
Mr Aso, a brash straight-talker, is seen as an antidote to the reserved style of Mr Fukuda. If selected, Mr Aso would become Japan's first Catholic prime minister.
This is his fourth attempt to win the leadership of the LDP.
The other candidates include Yuriko Koike, who is seeking to be Japan's first female leader.
Former defence chief Shigeru Ishiba and young reformist Nobuteru Ishihara are also in the running.
A total of 527 LDP members can vote - 386 MPs and 141 members of regional chapters.