The calls patch constituents through to their members of Congress to allow them to urge lawmakers to end the shutdown, which began early Tuesday morning as the clock ran out on the government funding measure and the two chambers were unable to come to a compromise. 

"While you we're sleeping, Congressman Tim Walberg (R-Mich.) shut down the government. You heard that right. But even worse, Congressman Walberg is still getting paid, and he's just not listening to our frustration. All because of his demand to take away your benefits and project insurance company profits," the robocall charges. 

It then allows the listener to connect to the congressman to "tell him to end the shutdown, stop the nonsense and focus on common-sense solutions that protect our healthcare and grow our economy." 

The calls are targeting 63 House Republicans, a wide swath of the caucus, only a handful of whom are vulnerable. A DCCC aide said the committee is hitting so many lawmakers because "all could pay a steep price" for the shutdown. 

Democrats believe the shutdown could be a political winner for them in 2014 if Republicans get the blame, much as it was for Democrats during the 1995-1996 shutdown. Polling conducted before Monday has indicated Republicans will receive most of the blame for a shutdown.

But Republicans believe the fallout of ObamaCare implementation would be equally, if not more, detrimental to Democrats.

"Democrats have slept through the ObamaCare alarm, and the American people are paying the price. Rather than wake up to the dangers of ObamaCare, Democrats are hell bent on forcing ObamaCare on everyone in America but themselves," National Republican Congressional Committee spokeswoman Andrea Bozek said.

— This post was updated at 10:15 a.m. to reflect comment from the NRCC.