But the aide noted that the House is able to move quicker than the Senate, and this idea could come into play. If a decision were made to pursue this idea, then it would require Democratic support to pass in the House.
Senate Democrats meet with president
Senate Democrats met with Obama on Saturday, and a Senate Democratic leadership aide said the party is unified.
"Democrats are willing to negotiate on anything Republicans want to discuss as soon as we reopen the government and pay our bills," said the aide.
Another Democratic source said party leaders regard Republicans as lacking a coherent position. They hope McConnell can "cut through the clutter," the source told CNN's Dana Bash.
The sources, who are familiar with the talks, spoke on the condition of anonymity so they could speak candidly.
Even as he demanded a "clean" bill, Reid said he and McConnell are involved in "cordial" and "preliminary" discussions.
"I hope that our talking is some solace to the American people and the world," Reid said. He said McConnell had approached him. "This hasn't happened until now."