A 64-year-old woman who calls herself a fan of a late British prime minister, Margaret Thatcher, is emerging as the frontrunner to become the next leader of Japan.

An outspoken conservative who opposes same-sex marriage, Sanae Takaichi, a 20-year member of the Japanese parliament, or Diet, would be the first woman to lead Japan if she gets the nod over her male rivals.

First, however, Ms. Takaichi must win election Saturday as president of the long-ruling Liberal Democratic Party. To lead the LDP, she needs to win a majority of party members in the Diet and Japan’s 47 prefectures. Next, she must put together a coalition with one or two minor parties to win the requisite majority of the Diet’s lower house, which elects the prime minister.

The contest for the LDP presidency “is widely expected to be decided in a runoff between the top two contenders, as none of the five candidates appears certain of winning a majority in the first round of voting by lawmakers and rank-and-file members,” Japan’s national news agency, Kyodo, said on the eve of the election. 

Ms. Takaichi, Kyodo said, “is the public’s favorite pick,” but she faces strong competition from the agriculture minister, Shinjiro Koizumi, who, at 44, would be Japan’s youngest leader in the post-war era. He also is the heir to a political dynasty: His father, Junichiro Koizumi, was prime minister between 2001 and 2006.

Kyodo said Mr. Koizumi “is the most popular among fellow lawmakers,” while the chief cabinet secretary, Yoshimasa Hayashi, “a dovish moderate with extensive ministerial experience, is rapidly catching up.”

That Japan is looking for a new prime minister shows the weakness of LDP rule among voters disturbed by government corruption along with rising prices.  The current prime minister, Shigeru Ishiba, in office for barely a year, said he would step down after the LDP lost its majority in the Diet in elections that he had called to show the LDP’s popularity and head off discontent.

Whoever winds up on top will get to summit with President Trump as he makes a stop at Tokyo on October 27 — that is, assuming someone has won a majority in the Diet. The meeting will likely echo Mr. Trump’s meeting at the White House with Mr. Ishiba in February, when they reaffirmed the strength of the alliance and promised to work on cutting down Japan’s enormous trade surplus with America.

This time around, though, the verbiage may be more strained, as Japan squirms under the 15 percent tariffs imposed by Mr. Trump, moves closer to America’s other northeast Asian ally, South Korea which is suffering under similar tariffs. Japanese also worry how much Mr. Trump will demand Japan pay toward the costs of American bases and forces in Japan while China and North Korea pose bigger threats than ever.

Japanese, like Koreans, fear Mr. Trump will want to scale down America’s commitment — about 50,000 troops, including a Marine division, a huge air base in the southernmost prefecture of Okinawa, another key air base at Tokyo, and a naval base on Tokyo Bay.

From Japan, Mr. Trump goes to the historic southeastern South Korean city of Gyeongju, renowned for ancient temples and palaces, for a gathering of leaders of the 21 economies lumped together into the Asia-Pacific Economic Cooperation group.  

Japan’s new prime minister should be at Gyeongju too — along with Communist China’s president, Xi Jinping, and South Korea’s president, Lee Jae-myung. As a sideshow, Mr. Trump might even fly up to the truce village of Panmunjom, on the North-South Korean line, and see his old buddy, North Korea’s leader, Kim Jong-un.

Risky though it is to make predictions, a poll conducted by the conservative newspaper Sankei Shimbun gave the impression that Ms. Takaichi will prevail. The poll, conducted at Kyoto University, showed that support for the LDP “would likely recover under Sanae Takaichi” but decline under the other two leading candidates.