Its choice is Sohei Miyashita, a former finance-ministry official, or Yoshiro Moni, a former cabinet-member.
LDP, Mr Obuchi's successor, Yoshiro Mori, won the job only after Mr Takeshita had given his approval.
Prime Minister Yoshiro Mori faces a no-confidence vote in the Diet today -- called by his own party.
So have Taku Yamasaki, boss of the party's policy research council, and Yoshiro Mori, chairman of its executive council.
Even his political godfather, Yoshiro Mori, a former prime minister, seemed at the weekend to give up on Mr Abe.
Japanese Prime Minister Mori Yoshiro lauded his late predecessor, Obuchi Keizo, last week at a memorial service studded with world leaders.
The day before Abe called on Obama, on February 21, former Japanese prime minister Mori Yoshiro was received by Putin in the Kremlin.
To general surprise, Junichiro Koizumi, billed as a reformer and a maverick, became prime minister of Japan after his hapless predecessor, Yoshiro Mori, resigned.
The prime minister, Yoshiro Mori, an old-style dispenser of pork with few obvious modern political skills, wins the approval of just 20% of opinion-poll respondents.
Yoshiro Mori, the prime minister, gets more unpopular by the day.
Public support for Mr Abe's government looks set to continue falling towards the nadir seen by Yoshiro Mori, prime minister in 2000-2001 (and Mr Abe's political godfather).
ECONOMIST: Another blow to Japan's Shinzo Abe as a minister goes
At the time of the cabinet reshuffle, Yoshiro Mori, one of the LDP's grand old men, foisted Taro Aso onto Mr Fukuda as the party's new secretary-general.
The following day, the Liberal Democratic Party (LDP) secretary-general, Mori Yoshiro, also 62, was elected prime minister by the Diet, securing 335 of 488 valid lower house votes.
Yoshiro Mori, Japan's much-abused prime minister, has been less fortunate.
Under Mr Fukuda, power has flowed back to the factions and to the party gerontocrats notably two former prime ministers, Yasuhiro Nakasone (89) and Yoshiro Mori (70) as well as Tsuneo Watanabe (81), publisher of Japan's biggest-circulation newspaper, the Yomiuri Shimbun.
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