The Asian country's state bank now holds a stake in GM Korea, which is GM's largest offshore operation.
GM's decision to close it Gunsan plant was the latest in a series of steps it has made to put profitability and innovation ahead of sales and volume.
At one time, Korea served as a major export base for GM, supplying a variety of small cars to both North America and Europe, among other markets.
Paik said the South Korean government needed reassurance from GM on its long-term commitment to the country before it can commit funds. The proposal is on top of a more than $2.2 billion debt-for-equity swap GM is offering to get financial support and tax benefits from Seoul.
The five trade pacts open South Korea to key Central American countries after its deals with the US, the European Union and China helped boost exports.
Last week, GM announced its plan to shut down the plant in Gunsan by May.
On Tuesday, Barry Engle, head of GM's global operations, met with a government-appointed task force in Bupyeong and told reporters the company wanted to stay in South Korea. "I'm encouraged by the discussions and I am optimistic that that is an outcome that together we can achieve".
In return, the Detroit automaker is seeking financial assistance from the South Korean government and is expected to also seek some aid from its traditionally militant unions.
General Motors (GM) has proposed a $2.8 billion fresh investment in South Korea over the next 10 years as part of its plan to restructure its embattled Korean unit, according to a South Korean senior government official.More news: Kamal Haasan launches his political party, names it Makkal Needhi Maiam
Engle, president of GM International, did not elaborate on whether GM's plan for the two new vehicle models were dependent on government support for the automaker, said Kim Sung-tae who attended the meeting.
It ran a surplus of almost US$1.9 billion (RM5.6 billion) with the the five Central American countries past year, according to figures from South Korean trade body Kita, with exports of US$2.2 billion and imports worth US$333 million.
The trade minister also stated that taxpayers' money would not be wasted in government efforts associated to the GM issue.
South Korea Trade Minister Kim Hyun-chong has signed a free trade deal for the nation with Costa Rica, El Salvador, Honduras, Nicaragua, and Panama.
"We need to have a closer look through the audit", the official said.
South Korean President Moon Jae-in called on Monday for a "stern" response to new U.S. tariffs on the South's exports as concern grew over looming trade restrictions by Washington.
Officials from the government and KDB have said that the automaker has not shared sufficient information about its finances or the cause of its mounting losses in the country. When GM left other markets, they did so boldly. GM wants to keep the remaining three plants - two in Bupyeong, just west of the capital city, and one in Changwon in the country's southeast - by allocating a new crossover vehicle for production here, the spokesman said.
But the automaker's decision to exit other unprofitable markets have exacerbated problems for GM Korea, which used to build numerous Chevrolet models GM once offered in Europe.