Analysts feared that the potential arrest would thrust the country's top car-maker into a leadership vacuum and derail its ambitions to become the world's fifth biggest car-maker by 2010 along with its affiliate, Kia Motors Corp.
Shares of Hyundai Motor fell 3% after the announcement, although South Korea's finance minister tried to calm investors by saying that he expected only limited impact from the case on the economy.
As well as seeking an arrest warrant for Chung Mong-koo, the company's chairman, prosecutors also seek to indict but not detain Chung Eui-sun, the chairman's son and president of Kia, senior prosecutor Chae Dong-wook told a briefing on Thursday.
The investigation has touched on how the country's sprawling family-run conglomerates, or chaebol, shift money within group companies, using complex share ownership networks to control their businesses.
Both executives were questioned in the past week over allegations that the Hyundai group operated slush funds and offered cash for political favours via a lobbyist.
"We have determined to arrest Chung on charges that he made a 100 billion won ($105.7 million) slush fund and caused 300 billion won worth of damage to the company through breach of trust," said Chae, referring to Chung Mong-koo.
He did not elaborate on allegations against Chung Eui-sun, but said more executives from Hyundai could be arrested.
Prosecutors have questioned Kim Dong-jim, the Hyundai Motor vice-chairman, who received a suspended two-year jail term in 2004 for playing a key role in raising a slush fund to support politicians during the 2002 presidential election.
A Hyundai Motor spokesman declined to comment on the case.
The company said in a filing to the stock exchange that it would not release its first quarter-earnings as planned on Thursday and did not give a date.
Combined exports by Hyundai Motor and Kia account for nearly 10% of total exports in South Korea, Asia's fourth-biggest economy, according to company data.
Han Duck-soo, the South Korean finance minister, told a weekly briefing: "We think that it will inevitably have some impact on the economy, but it won't be that huge."
Chae, a senior prosecutor at the Supreme Prosecutors' Office, said national interest would be kept in mind when meting out potential punishments over the case.
Last week, Hyundai, which makes the popular Sonata sedans, apologised and said its chairman's family would donate their entire stake in car shipping affiliate Glovis, worth $1 billion, to charity.