Cardin also said federal workers shouldn't suffer from the actions of Republicans, who he blamed for the shutdown.

"They are dedicated public servants who simply want to do their jobs on behalf of the American people," he said Tuesday. "They shouldn't suffer because so-called Tea Party Republicans, mostly in the House of Representatives, suffer from the delusion that shutting down the Federal Government will somehow prevent the Affordable Care Act from being implemented."

Cardin said federal workers have already suffered enough, after being hit by a pay freeze that has forced these workers to help reduce the deficit by $90 billion.

"On top of the pay freeze, hundreds of thousands of Federal workers have been furloughed because of sequestration," he added. "Their pay hasn't just been frozen; it has been cut."

The bill was introduced during the first day of the government shutdown — that shutdown was expected to continue for at least two days, as House Republicans had made no sign of agreeing to a continuing spending resolution without language that somehow undermines ObamaCare.

Similarly, Senate Democrats indicated they would continue to oppose any solution other than a "clean" spending resolution.

On Monday, a group of House Republicans and Democrats proposed a bill that would ensure pay for workers affected by the shutdown. The three Republicans on that bill are from Virginia, where thousands of affected government workers live.