Former Prime Minister Kim Boo-kyum was the co-chair of the Democratic Party of Korea's campaign committee in the April 10 general elections. Yonhap

The presidential office is struggling to find a new prime minister as potential candidates seem reluctant to take on the post following the election defeat of the ruling party.

Prime Minister Han Duck-soo offered to resign, Thursday, a day after the April 10 general elections to take responsibility for the People Power Party's crushing defeat.

President Yoon Suk Yeol has not approved Han's resignation as of Tuesday noon, but the presidential office is screening possible candidates since the election result has severely weakened the country's conservative president.

There are growing calls that Yoon should appoint the next prime minister from the opposition party or pick a moderate figure with bipartisan appeal in order to bring unity. Appointing a prime minister requires approval from the National Assembly — meaning approval from the main opposition Democratic Party of Korea (DPK) that has held the majority during the current 21st Assembly and will continue to do so during the next 22nd Assembly.

Whoever takes the reigns faces a myriad of challenges including diminishing competition between political parties and political division. The next prime minister is likely to face DPK-led opposition against a push for policies that require legislative endorsement

Former Prime Minister Kim Boo-kyum's name first surfaced publicly after lawyer Shin Pyong, known to be a mentor for President Yoon during his presidential election campaign, said Kim is willing to become the next prime minister if there is consent from the DPK. There was a flurry of media interest this week on whether Kim could take on the post.

However, Kim expressed discontent over the reports, saying, "What's more important right now is not who gets appointed as the next prime minister. President Yoon should meet with DPK Chairman Lee Jae-myung and propose his comprehensive reform plan."

Rep. Kwon Young-se attends a meeting of ruling People Power Party lawmakers that have been elected more than four times at the National Assembly in Seoul, Monday. Joint Press Corps

Many see Rep. Kwon Young-se of the PPP, who also clinched his fifth parliamentary term, as a potential candidate for the job.

Kwon is categorized as a practical and moderate politician, due to his experiences of arbitrating factional infighting within the conservative bloc. Because he served as the unification minister until July last year and kept his distance from mudslinging partisan fights, political observers describe him as a balanced candidate.

However, Kwon played down reports, saying, "I think that's a groundless rumor. Right now, I'm not even thinking about that."

Former Land Minister Won Hee-ryong, who lost to the DPK chairman in last week's elections, is also a potential candidate, but he is said to have opposed the suggestion.

Rep. Ko Min-jung of the DPK, a former spokesperson for former President Moon Jae-in, said, Tuesday, the unwillingness of the candidates means Yoon is falling into a lame duck status.

"President Yoon needs someone whom he can share his burden with. There will be very few ruling party members who want to become the next prime minister because they all know the president was responsible for the election defeat," she said on an MBC radio show. "This signals Yoon's lame duck status."

Other frequently mentioned candidates are Kim Han-gil, chairman of the Presidential Committee of National Cohesion, Rep. Joo Ho-young, who secured his sixth term as a lawmaker, and former Vice Assembly Speaker Park Joo-sun.