While the public outcry over the incident has not yet grown to a level that could jeopardise President Lee Jae-myung or his party's majority in parliament, some protesters have accused Lee's liberals of playing a part in what they called "election fraud".
Lee has said the issue exposed by the incident was a matter of democratic integrity, not a conspiracy to rig an election.
He met the speaker of parliament and the heads of the Supreme Court and the Constitutional Court, as well as Prime Minister Kim Min-seok, to discuss what they said was "a grave infringement of voter rights" and agreed a speedy and thorough change was needed, Lee's office said in a statement.
"The president asked the officials to do everything they can to help overcome this crisis in our constitutional order," it said on Monday.
Earlier, Lee said he was shocked and dismayed by the June 3 incident that kept some voters waiting for hours at polling stations for ballot papers to be delivered.
The incident was a serious blow to the country's reputation as a model democracy and protests questioning the fairness of the process were entirely justified, Lee told a news conference on Monday.
"It was just ridiculous," Lee said.
"It was probably something that's hard for even people in a lesser developed democracy to imagine."
Thousands of people have protested outside a ballot counting station in Seoul, demanding a new election.
Lee said at the Blue House meeting that he supported the protests and wanted to thank them for raising the issue.
The head of the NEC, an independent body that oversees all elections in the country, has resigned to take responsibility.
The commission has said it printed ballot papers for 73 per cent of the total eligible voters - calculated based on previous turnout rates - and polling stations in some districts were slow to receive additional ballot papers after they started to run out.
Lee has ordered a thorough investigation into the ballot paper shortage, and the ruling Democratic Party said it plans to launch a national inquiry.
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