But Joel Kaplan, the White House deputy chief of staff, said there was still "a lot of drafting going on" as the parties worked out the final details.
"We made very substantial progress and have a good conceptual agreement," he said.
He said a final bill would ask car makers who receive government financing to face "fundamental restructuring or bankruptcy".
The bailout is designed to allow General Motors and Chrysler to avert threatened bankruptcy with short-term loans until March, when president-elect Barack Obama's new administration is likely to agree more funding.
Ford is not requesting immediate help but is asking for a line of credit in case its finances worsen.
Democratic and White House negotiators agreed last week that the money would come from an Energy Department fund established to help Detroit make more fuel-efficient cars, not from the recent $700bn financial bailout, Reuters reported.
Democrats had pushed for the auto firms to receive funds from the bailout passed by US congress last month, but the Bush administration has resisted such attempts, saying the money should come from the energy department's $25bn loan programme.
The measure would grant loans to all three firms but also demand that the industry restructure itself in order to survive.
Car sales have slumped amid an economic
It would also put a "car tsar" chosen by George Bush, the US president, in charge of monitoring the bailout, AP said.
Mitch McConnell, the senate Republican leader, said his party's senators wanted to examine the pending bailout bill before making any commitments.
McConnell said Republicans want a bill that forces car makers, who have struggled to compete with more fuel-efficient foreign-made vehicles, "to reform their bad habits".
Al Jazeera's Rob Reynolds said many Republicans had "nothing to lose" as some were coming to end of their congressional terms.
However others were opposed to the deal as they were against any intervention in the US's free market system, our correspondent adds.
Democrats have also demanded, against White House opposition, that car makers end legal action against states seeking to reduce pollution.
The deal is not as large as the $34bn the chief executives of all three firms had asked for from congress last week and the rescue plan will most likely also face opposition from Republicans in congress.