The union would initially receive a 17.5 percent stake to finance a health care trust for its retirees. It has also received warrants to raise that holding to 20 percent  but those warrants are exercisable only if new G.M.’s value hits $75 billion.

Once the union and bondholders achieve their full stakes, the government’s share would drop to 55 percent.

The hope is to create a new G.M. by late August, people with knowledge of the matter said.

During a briefing on Thursday, administration officials said they expected that G.M. would emerge from bankruptcy in 60 to 90 days, but would probably not be a publicly traded company until sometime later, possibly after a public offering. Until then, there would not be a ready market for the equity holdings of former bondholders and others.

The administration officials said that G.M.’s balance sheet going forward could make a $15 billion market capitalization possible within a relatively short period of time, but did not offer details on how its value would rise to that level. The current market capitalization, said an administration official close to the negotiations, reflected a capital structure that was unacceptable.

G.M.’s current market capitalization is about $683.8 million; it was close to $22 billion a few years ago. (By comparison, Toyota is now worth $123 billion, and its market capitalization was $200 billion in 2006.)

G.M. bondholders rejected the initial offer because they were upset that the U.A.W.’s health care trust, to which G.M. owes $20 billion, received a larger stake than the debt holders, who were owed $27 billion.

Under the proposal, bondholders conceivably will outrank the health care trust, once the warrants are exercised. Not only does that soothe any ruffled feelings, but it will create good will with the lenders G.M. will need to tap after it emerges from bankruptcy. On the other hand, the union will hold debt and preferred stock that helps guarantee its health care trust will be financed even if the new company falters.