By Linda Sieg

TOKYO (Reuters) - Japanese Prime Minister Shinzo Abe's political crisis deepened on Monday after polls showed that suspected cronyism scandals have pushed his support to record lows and a popular predecessor said Abe would probably resign in June.

In another headache for the conservative premier, the finance ministry's top bureaucrat has come under fire after a weekly magazine reported he had sexually harassed several female journalists. The bureaucrat on Monday denied the accusations and said he would file a lawsuit against the magazine's publisher.

Abe’s sliding ratings raise doubts about whether he can win a third three-year term as ruling Liberal Democratic Party (LDP) leader in a September vote, that he needs to win to stay in office, or whether he might resign before the party vote.

Speculation has even emerged that Abe, who surged back to power for a second term as prime minister in December 2012, promising to reboot a stale economy and bolster Japan's defenses, could call a snap general election as he did last October when his ratings were in a similar slump.

A survey by broadcaster Nippon TV released on Sunday showed Abe's support had sunk to 26.7 percent, the lowest since he took office in December 2012. An Asahi newspaper poll on Monday put his support at 31 percent.

The latest signs of trouble for Abe come ahead of a summit this week with U.S. President Donald Trump, where the difficult topics of North Korea's nuclear and missile threats and trade will be on the agenda.

"The situation is getting dangerous. Won't Mr Abe resign around the time parliament's session ends?" former premier Junichiro Koizumi, a one-time Abe mentor, said in an interview published by the online edition of the weekly magazine Aera.

The session ends on June 20.

Koizumi - a critic of Abe's support for nuclear power after the 2011 Fukushima accident - said that if Abe hung on, it could hurt LDP candidates in an upper house election next year.

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Crowds of protesters demonstrated near parliament on Saturday, holding signs saying "Abe is Over" and chanting "Abe quit!" Organisers said 50,000 people took part.

The Asahi survey, in line with others, showed that two-thirds of voters did not trust Abe's explanations that he was not involved in the cronyism scandals.

Abe has denied that he had intervened to ensure preferential treatment for the educational institution Kake Gakuen, run by his friend Kotaro Kake, to set up a veterinary school.

He has also repeatedly denied that he or his wife intervened in a heavily discounted sale of state-owned land to another school operator, Moritomo Gakuen, which has ties to his wife.

FINANCIAL MARKETS BLASE

The Moritomo affair has ensnared the finance ministry, which has admitted officials doctored documents related to the land-sale.

The ministry is also in an unwelcome spotlight over the sexual harassment allegations against Administrative Vice Finance Minister Junichi Fukuda.

He has denied the allegations, published by the Shincho magazine, and vowed to sue its publisher for defamation.

Fukuda apologized for causing trouble for the minister and ministry officials "by inviting public distrust", the ministry said.

Financial markets have not reacted to the latest fall in Abe’s support as many investors think he can survive, as he did last year when the cronyism scandals eroded his ratings.

"If Abe will have to leave, that would be a big deal but I don’t see that happening yet,” said Soichiro Monji, chief strategist at Daiwa SB Investments.

Many investors also noted the risk of “Abexit” looks small compared with the more immediate risks of a possible trade war and a planned summit between Trump and North Korean leader Kim Jong Un that could leave Japan isolated.

Who would replace Abe, who has managed the rare feat of staying in office more than five years, is unclear.

Former cabinet minister Shigeru Ishiba, who wants to challenge Abe for the top post, topped a list of politicians that respondents to a weekend Kyodo news agency survey saw as best suited to become the next premier, with 26.6 percent.

Popular young LDP lawmaker Shinjiro Koizumi - ex-prime minister Koizumi's son - ranked second with 25.2 percent, with Abe in third place with 18.3 percent.

Former foreign minister Fumio Kishida, another possible contender, was fourth with 5.9 percent followed by Internal Affairs Minister Seiko Noda at 3.6 percent and Foreign Minister Taro Kono with 2.9 percent.

Among LDP supporters, however, Abe was top with 36.7 percent against 24.7 percent for Ishiba, Kyodo said.

None of the potential successors has outlined clear agendas, although Ishiba and Kishida have suggested more attention should be paid to public debt and that the hyper-easy monetary policy central to the premier's "Abenomics" growth strategy could not continue forever.

(Additional reporting by Hideyuki Sano and Tetsushi Kajimoto; Editing by Michael Perry, Robert Birsel)