The offer of resignations came after President Roh Moo-hyun tendered a public apology for the involvement of a close aide in a political funding scandal and decided to seek a fresh vote of confidence in his administration.
"All the 18 cabinet members decided to submit resignations holding themselves responsible for the situation," a spokesman for the Prime Minister Goh Kun said.
But a South Korean television station quoted the president as saying he would not accept the resignations.
Together with the cabinet, the Presidential secretary and other secretaries had also offered to resign.
President Roh said he felt a shared responsibility for the scandal, involving his close aide and the SK Group conglomerate.
He pledged to seek renewed support from the public before next year's parliamentary elections.