Aso signalled on Tuesday that he planned to run and thought he was a suitable candidate.
"I think [Fukuda] felt he had work that was left undone, and he said he wanted it to be carried out," Aso told a news conference.
"As someone who discussed these issues with him, including the economic package, I think I have the credentials to take that on," said the veteran legislator, currently secretary-general of the LDP.
But analysts said the next LDP prime minister would face similar woes given the parliamentary deadlock and the party's rusting political machine and scandal-tainted image.
|Analysts have warn Fukuda's departure could cloud Japan's reform outlook [AFP]
The main opposition Democratic party has made no secret of its desire to oust the LDP and its junior coalition partner, the New Komeito.
Talk of a broad realignment of party alliances has been simmering since the opposition took control of the upper house last year, but it remains to be seem if early moves pick up steam.
Fukuda, 72, had been struggling to cope with a divided parliament where opposition parties have the power to delay legislation, and his sudden exit raised questions about his conservative party's ability to cling to power or even hold together after ruling Japan for most of the past 53 years.
The moderate conservative who favoured close ties with Japan's Asian neighbours, only took office last September after his predecessor, Shinzo Abe, similarly quit suddenly after just a year in office.
Fukuda's resignation does not automatically mean an election being held earlier than the due date in September next year.
But whoever the LDP picks as its leader, and thus the next prime minister, might choose an early poll to take advantage of any rise in public support.
A complete deadlock in parliament could also force the prime minister to call early polls.
If the LDP goes past the outspoken Aso again, other potential leadership contenders are Kaoru Yosano, the economics minister known for his commitment to fixing Japan's tattered finances, and conservative Yuriko Koike, who was briefly last year the country's first female defence minister.
Seiko Noda, the minister in charge of consumer affairs, has also been mentioned by the media as a possible successor to freshen up the LDP's image.
Markets on Tuesday appeared to take Fukuda's departure in their stride, with the Nikkei stock average and government bond futures rising after initial wobbles.
But analysts said the markets would probably sag as Fukuda's departure was expected to cloud the reform outlook and potentially spur debt-funded spending, warning that Aso might try to spend Japan's way out of its economic woes.