To win, a candidate needs to take a majority of the 527 votes to be cast, 386 cast by party members in the Japanese parliament and 141 by members of regional chapters.
Aso has already won 90 votes from regional chapters, according to polls by Yomiuri Shimbun newspaper and other media.
Polls have also suggested about 60 per cent of LDP members of parliament would vote for him.
Aso, a Catholic with a reputation as an outspoken conservative, has vowed to turn around Japan's sagging economy and defend its national interests by strengthening Japan's alliance with the US.
He is seen as a stark contrast to Fukuda, who was reluctant to engage with the opposition and was seen as having problems connecting with voters.
But his opponents have criticised his plans to boost the economy by stepping up government spending.
Whoever wins is widely expected to call snap general elections - possibly for as early as late October - in the hope of thwarting an increasingly popular opposition.
Last year the LDP, which has ruled Japan almost continuously since 1955, lost control of the upper house of parliament for the first time.