The scandal was triggered by the suicide of a powerful businessman, who before his death alleged he had given political funds to the prime minister and two close Park associates.
The president has vowed to punish "anyone" found involved in corruption, while Lee has ignored calls from the opposition New Politics Alliance for Democracy (NPAD) to step down.
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"Our party will push for a bill to dismiss him," Moon said on television.
He urged the ruling Saenuri Party to agree on a quick vote in parliament.
Sung Wan-Jong, the former head of a bankrupt construction company, hanged himself on a hillside near his house in Seoul on April 9.
In his pocket, investigators found a note that listed the names of eight people -- including Lee and presidential chief of staff Lee Byung-Kee -- alongside figures that were alleged to indicate bribery sums.
The suicide came as Sung was about to be questioned by prosecutors over allegations that he created a slush fund with embezzled company money to bribe politicians and government officials.
In an earlier newspaper interview, Sung claimed to have given Lee 30 million won ($27,000) in cash in 2013 when he was running for a parliamentary seat.
Lee has angrily denied the allegation and said he would "lay down" his life if prosecutors find any evidence that he took bribes.
The affair is a fresh blow to President Park, whose popularity ratings were only just beginning to recover from the hit they took after the Sewol ferry disaster a year ago that killed 304 people.
Park has repeatedly been criticised for the appointments she has made in the two years since taking office.
A number of her nominees for senior posts have been forced to withdraw because of allegations of past misconduct.
The prime minister is a largely symbolic post in South Korea where power is concentrated in the executive.
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