WASHINGTON — The fate of the “cash for clunkers” program remained uncertain on Monday even as sales figures from automakers demonstrated that people had flocked to dealers to trade in old gas guzzlers.

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The White House urged the Senate to add $2 billion to the program, as the House voted to do last Friday before leaving for its August recess. Still, dealers around the country stopped promising the rebates to car shoppers on Monday, because of uncertainty about how much of the $1 billion initially allocated had been used up, or when or whether more money would be available. The Senate begins its recess this Friday.

The short-term tonic of the first billion dollars was evident, though, in sales figures...