“There’s certain things that we have to do. And I don’t have any doubt that we can work with each other,” McConnell said in a brief interview. “Her problem at the moment is internal [in her caucus], not with us, but how to manage that whole process and how can we fit other things in it.”

Pelosi spokesman Drew Hammill shot back: “The majority leader knows as much about what goes on in the Democratic Caucus as he does about passing legislation: absolutely nothing.”

Pelosi and McConnell have barely spoken recently about funding the government, according to multiple people familiar with their interactions. Instead, the No. 2 House Democrat, Majority Leader Steny Hoyer, called McConnell on Monday to talk spending bills.

“McConnell wants to get these bills done, and that's fine. I want to get the bills done,” Hoyer told reporters Tuesday. “There’s no reason whatsoever from a process standpoint that we can’t do the appropriations process by” the next deadline.

“Right now, the president seems to want to see this branch as chaotic as his branch. And I worry that McConnell will play into that."

People close to the speaker say her relationship with McConnell soured after the border funding debacle over the summer. McConnell refused to bring up a House-passed border aid package that included additional protections for migrant children, forcing Pelosi to take up a Senate bill that lacked those restrictions.

These days, McConnell said “there’s no particular reason” to talk with Pelosi about funding the government, noting he and the speaker recently worked closely to pass a large-scale budget deal intended to ease the threat of a government shutdown.

“I don’t have any advice to give her on [her internal politics], except I wish she’d pass USMCA and we could at least rack up a victory,” McConnell said in the interview, referring to the U.S.-Mexico-Canada trade agreement.

The two longtime party leaders are recognized even by their adversaries as consummate dealmakers, able to separate political beefs from the must-do business of Congress. Their manners are also diametrically opposed: At a White House meeting last week Pelosi was seen standing and chastising the president after he called her a third-rate politician; McConnell didn’t speak in the meeting or afterward about the blow-up.

The impeachment drive will test whether the two can stave off legislative disaster as the House seeks to remove a president that McConnell’s eager to protect and Pelosi openly loathes. The next funding deadline is likely to come at a politically inopportune time — as the House is moving rapidly toward an impeachment vote and when bipartisan cooperation will be in short supply.

House leaders have said they want to move as expeditiously as possible in their investigation into President Donald Trump, with lawmakers privately speculating a vote on articles of impeachment will happen by the end of the year if not sooner. McConnell has told Republicans to prepare for a December trial, possibly ending around Christmas.

And Trump remains the X-factor. Even if Pelosi and McConnell can reach a deal, the president must go along with it, or the government will shut down once again.

“Right now, the president seems to want to see this branch as chaotic as his branch. And I worry that McConnell will play into that,” said Rep. Dan Kildee (D-Mich.).

Almost everyone in the Capitol acknowledges that the prospects for passing ambitious legislation like a new North American trade deal, background checks on gun sales and prescription drug price reform have taken a major hit over the past month as the impeachment push cranked up. Yet it’s the routine business of funding the government, as well as passing an annual defense bill that will prove the real measure of Washington’s dysfunction.

“They’re both professionals, right? I don’t think either one of them are the type of people that let political emotions get in the way of doing what they think is in their caucuses’ best interest,” said Sen. Chris Murphy (D-Conn.). “I don’t think Donald Trump can, but he’s not going to be in the room.”

Others aren’t as optimistic.

Pelosi and McConnell have less than a month to fund the government past Nov. 21 and until the end of the year to pass the National Defense Authorization Act, one of Congress’ few annual legislative successes. And with lawmakers projecting that impeachment will suck up most of November and December, it’s not hard to see the pile-up coming.

“We only have so much bandwidth. And if the bandwidth is going to be all used up pursuing this futile effort to remove the president, that’s going to come at a cost,” said Sen. John Cornyn (R-Texas), who is close to McConnell. “Hopefully, [government funding and defense bills] won’t be casualties. But around here, it’s anybody’s guess.”

Lawmakers are already talking openly about the likelihood that Congress delays a final funding agreement and passes some kind of stopgap bill to keep the government open until early next year, when the impeachment inquiry is finished.

Even if Pelosi and McConnell came together in the next three weeks, passing long-term spending legislation would be a huge lift. A best-case scenario, laid out by Republicans, is to again fund the majority of the government and isolate key funding disputes on perhaps the wall and Department of Homeland Security.

“You still may be dealing with Homeland Security and the wall issue,” said Senate Majority Whip John Thune (R-S.D.) “But it’s a much smaller universe.”

The Senate hasn’t passed any appropriations bills, although it advanced a package of four bills on Tuesday. But a vote on defense and labor spending could fail later this month over Democratic opposition to Trump’s move to shift military funds to build his border wall.

Congressional leaders haven’t even settled the topline spending levels for federal agencies, prompting Hoyer to reach out to McConnell this week to avoid a complete debacle like the historic 35-day shutdown earlier this year.

Meanwhile the two congressional leaders are going at it.

McConnell repeatedly called out Pelosi this week for not passing the UMSCA, accused her of bowing to the “far left,” and said “the only thing that seems to really inspire House Democrats these days is their obsession with overturning the results of the 2016 election.”

Pelosi recently alluded to McConnell as a turtle who rarely “comes out of his shell” during White House meetings and has begun calling him “Moscow Mitch,” a nickname that deeply irritates the Kentucky Republican.

And it’s going to only get uglier.

“I don’t know what their relationship is. I have to imagine it’s harder when you’ve got stuff like this going on,” said Sen. Lisa Murkowski (R-Alaska). “Much harder.”