However, even if the Cash for Clunkers bill were to make it to the Senate schedule this week, the program may be completely out of money. At that point, if any of the amendments expected to be offered pass, the newly-amended program would have to be reconciled with the bill that passed the House of Representatives who have already adjourned for August recess.
On C-SPAN Sunday, Transportation Secretary Ray LaHood said he had "great faith" that the Senate would pass the bill.
"We will continue the program until we see what the Senate does," he said. "I have great faith that the Senate will pass this."
The White House has assured dealers that they will be reimbursed for any deals closed over the weekend, but, in the meantime, car dealerships that continue to offer the government's incentive are doing it on faith.
In Friday remarks about the economy, Obama said the program "has succeeded well beyond our expectations and all expectations, and we're already seeing a dramatic increase in showroom traffic at local car dealers."
Citing unreleased National Highway Traffic Safety Administration data given to their staffs, Feinstein and Collins said consumers taking advantage of the clunkers program have bought more fuel efficient vehicles than they had to.
Feinstein and Collins, two original sponsors of the measure, were frustrated when its fuel economy requirements were watered down and last Friday demanded tougher fuel economy requirements in order to pass the legislation.
But they pointed to numbers given to their offices by the NHTSA, which showed that of the 120,000 sales so far, the average fuel economy for a purchased car was 25.4 mpg. For cars it was over 28 miles, for SUVs over 21 and for trucks over 16.
While 34 percent of consumers receiving the credits bought SUVs and minivans, Feinstein said 83 percent of the vehicles turned in were trucks and SUVs, and the average refund being given was $4,327. She said NHTSA expects another 130,000 cars can be bought through the program before it reaches $1 billion.
Collins said she had urged GOP leaders to work with Senate Majority Leader Harry Reid, D-Nev., to come to an agreement to allow a vote to occur this week. All senators must agree to the vote for it to happen. But with a summer recess looming, peer pressure against objecting could be high.
Sen. Charles Schumer, D-N.Y., who also appeared at the press conference with Collins and Feinstein, said he expected any vote on the clunkers bill passed by the House would require 60 votes.
"The bottom line is if it ain't broke, don't fix it," Schumer said, making full use of car metaphors. "It's working in every way. It's working as a stimulus, it's working to help families, it's working to improve mileage. What we need to do here is not put on the brakes, but step on the gas."
ABC News' Lindsey Ellerson contributed to this report.