Roh Moo-hyun made the offer to the head of the opposition Grand National Party (GNP) Choe Byung-yul and other senior party officials, according to the president’s office on Sunday.
Talk of resignation has dominated the local press for the second time in two months after details of the scandal leaked in October.
But Roh said the referendum he had proposed to hold this month was "not possible" and that he would seek another way to test his mandate.
He now assures political party leaders he will quit if the amount of illegal funds his camp received surpassed a fraction of those of his opponents.
"If the amount of our illegal funds is more than one-tenth of the Grand National Party's, I will resign from my presidential post and retire from politics."
Roh did not explain how he arrived at that formula.
When the scandal first broke, Prime Minister Goh Kun confirmed all 18 cabinet members had submitted their resignations.
However a South Korean television station quoted the president as saying he would not accept the resignations and would take personal responsibility.
Political opposition involved
Even though a former presidential adviser was arrested in October for taking 1.1 billion won ($926,600), the opposition party are as much involved.
"If the amount of our illegal funds is more than one-tenth of the Grand National Party's, I will resign from my presidential post and retire from politics"
South korean president
Prosecutors probing illegal campaign funds in the December 2002 presidential election say then opposition candidate Lee Hoi-chang's party took as much as 50 billion won ($42 million) in illicit donations from huge business conglomerates.
Surh Jeong-woo, a legal adviser to Lee who was arrested on Wednesday for taking illegal funds, admitted to taking 10 billion won from Hyundai Motor, the prosecution said.
The funds from the country's biggest carmaker were in addition to 15 billion won each from Samsung and LG that Surh admitted receiving on behalf of Lee.
Amazing cash handover
In one of the more spectacular revelations, prosecutors said an LG executive handed Surh the keys to a truck stacked full with 63 boxes containing the 15 billion won at a rest step on South Korea's main cross-country expressway.
The now retired Lee made a rare public appearance last month in which he apologised to the country and said he was willing to go to jail for the party's illicit actions.
The GNP has admitted receiving the cash, but insists that prosecutors are unfairly focusing only on its illegal fundraising, while ignoring problems in Roh's ruling camp.
The GNP recently used its majority in the 273-seat National Assembly to overturn a presidential veto and force Roh to appoint a special counsel to investigate his aides.
The 90-day special probe will open next month.