Reid's remarks came after a bruising defeat on Wednesday night for President Obama's jobs bill, which garnered the support of just half the Senate in a procedural vote. Several Democrats voting to bring the bill to the floor said they actually opposed the underlying legislation in its current shape.

No Republicans supported the $447 billion spending bill, but Reid said two GOP members had told him after the vote that they might support some elements of the bill if they were brought up alone. 

"I am looking forward to working with my Republican colleagues in moving parts of the bill they like," concluded Reid on Thursday morning.

Senate Minority Leader Mitch McConnellAddison (Mitch) Mitchell McConnellGOP senator: Democratic opposition to Pompeo 'driven 100 percent by politics' Pompeo lacks votes for positive vote on panel GOP poised to advance rules change to speed up Trump nominees MORE (R-Ky.), for his part, seemed open to considering the parts of the bill that cut taxes.

"Pieces of the second stimulus might well be appropriate," said McConnell in response to Reid's floor speech.