While in Louisville over the Senate's Memorial Day recess, McConnell said he was "still in favor of unemployment insurance," but the additional $600 granted under the CARES Act inhibited certain industries' efforts to bring back workers as the economy gradually opens.

"What I thought was a mistake was the bonus we added that small businesses all over the country are saying make it more lucrative to not work than to work. That's exactly the opposite of what we want to do," McConnell said. The $600 supplemental benefit is set to end July 31.

Democrats have not only insisted on the extension of that supplemental benefit, but some want to even further. An unnamed Senate aide told Politico that Senate Minority Leader Chuck Schumer (D-N.Y.) is considering a push for an "automatic stabilizer," which would tie unemployment benefits to the condition of the economy. Speaker Nancy Pelosi has also publicly endorsed the idea.

Both Republicans and Democrats have floated other options, many tied to a return to work.

Sen. Rob Portman (R-Ohio) has proposed providing workers with an additional $450 a week bonus on top of their current wages as an incentive to go back to work. White House economic advisor Larry Kudlow said recently on Fox News that Portman's plan was "something we're looking at very carefully."

On the other side of the aisle, Sen. Mark Warner (D-Va.) has raised the possibility of dramatically expanding the employee retention tax credit, an idea that has support from both moderates and liberals in the Democratic caucus—Schumer included. Warner also suggested in an interview that there could be "some collaboration" between his proposal and Portman's.

Rep. Dean Phillips (D-Minn.) believes the question of renewing aid is "an example of where there are two truths. One truth is that, yes, the $600 amplification is going to complicate things for many businesses to re-attract their employees. That is a fair assessment. But the other truth is that we have a problem in our country with people struggling to put food on their tables and a roof over their head."

Glimmers of bipartisanship aside, some senators are settling in for a long fight. "I don't expect we'll see any immediate action in the Senate," majority whip Sen. John Thune (R-S.D.) said at a recent online event. The gridlock is familiar but no less frustrating to observers like Meredith McGehee, executive director at Issue One, who told Reuters that Congress is "playing a dangerous game of chicken."

Read more: 

• Unemployment claims fall as states begin to reopen
• Middle-income retirement participants 'hit from both sides' by COVID-19
• Study: Nearly half of adults say their family has lost work or income amid the pandemic