Senator Chuck Schumer was downright effusive when asked about Senator Mitch McConnell’s health status after the Kentucky Republican’s second frozen moment before television cameras last week.

“I’m very glad to see McConnell back,” Mr. Schumer, the New York Democrat and majority leader, told reporters on Wednesday.

It was more than just the standard best wishes for a Senate colleague facing health concerns. Though Mr. McConnell, 81, the longtime Republican leader, has been a leading nemesis of congressional Democrats for decades, they happen to share some common interests at the moment, with a government shutdown looming and clashes ahead on aid to Ukraine, disaster recovery spending and a potential impeachment of President Biden.

So despite their often hostile history with Mr. McConnell, some Democrats privately concede they are anxious about the prospect of his abrupt exit. They are counting on Mr. McConnell’s skills as a legislative tactician, his political stature and his general wiliness to help navigate congressional confrontation as they face a season of clashes with House Republicans driven by their party’s most far-right element.

“He is the devil they know and a very pragmatic leader,” said Senator Kevin Cramer, Republican of North Dakota, about Mr. McConnell and Senate Democrats. “And there is no one more shrewd.”

Mr. McConnell is a stalwart backer of maintaining financial assistance to Ukraine despite growing reservations in his own party, and he has the influence and record to keep a majority of his fellow Senate Republicans on board. He has also made it very clear that he wants to avoid a government shutdown and shares the view of Senate Democrats that federal spending has to be higher than what House Republicans are pursuing. And he has suggested that initiating an impeachment of Mr. Biden is not a great idea.

Democrats who often rip Mr. McConnell are refraining from doing so right now.

“There are extremes, mainly in the House, that I worry about, but I think he speaks for the mainstream Republican Party on most issues,” said Senator Richard J. Durbin of Illinois, the No. 2 Democrat, who had a lengthy conversation on the floor with Mr. McConnell about his health when the Senate convened Tuesday after a long summer recess. “He can play an important role in his own ranks in helping us come up with a bipartisan solution to our challenges.”

Mr. Biden, who has a track record of negotiating with Mr. McConnell, was among the first to reach out to the Republican leader last week after the episode in Kentucky and later offered public assurances about his condition. Officials in the Biden White House see Mr. McConnell as a figure they can depend on to avert a crisis, even though he might hold on to his cards until the last possible second.

It is a sensitive situation for both sides. Mr. McConnell is not particularly keen to be caught working hand in hand with Democrats, further infuriating his far-right critics, who already see him as a charter member of the swamp and what some of them call the “uniparty” mind-set in Washington. And Democrats do not want to be seen as cozying up to Mr. McConnell after their experiences dealing with him in a long string of toxic confrontations, such as when he blocked President Barack Obama from filling a Supreme Court seat for a year, keeping it open until it could be filled by Donald J. Trump.

They know more disagreements lie ahead.

One Democrat said privately that they are hardly “buddy-buddy” with their usual foe. Still, there is an acknowledgment that on some of the big issues they are facing, he can be more of an ally than an obstacle. Mr. Schumer said on Wednesday that “we’re working well together.”

“On some things, Leader McConnell and I disagree vehemently,” Mr. Schumer said. “On others, we agree and work together to get done.”

On Wednesday, Mr. McConnell both publicly and privately made a push for $24 billion in new aid for Ukraine while conceding there was “a difference of opinion in my party” on the money sought by the Biden administration. To try to strengthen his case, he noted that much of the money is being spent in the United States on munitions and other supplies, and that Ukraine is doing the fighting so Americans do not have to. Being Mitch McConnell, he did take time to fault the administration for the way it is pursuing its Ukraine policy, but he said he would stick with the president.

“I think he could have done it more skillfully, but he is supporting the effort, and I intend to continue to support it, and I hope the majority of my colleagues will feel the same way,” Mr. McConnell said.

Mr. McConnell also took the opportunity on Thursday to bash one of the administration’s and congressional Democrats’ prized new initiatives — negotiating lower Medicare prescription drug prices — as “prescription drug socialism” in a reminder that he is hardly on board with everything, or even most things, coming out of the White House.

What Mr. McConnell has consistently been on board with, even as a growing segment of his party has come to disdain it in recent years, is the steady functioning of the federal government.

In recent weeks, he has been strongly supportive of the bipartisan work of the Senate Appropriations Committee to fund government agencies at the level agreed to in the deal Mr. Biden reached with Speaker Kevin McCarthy to suspend the debt limit. He has clearly stated that the lower levels sought by House Republicans are not going to fly in the Senate. Even as some among the far right in the House suggest that a government shutdown after funding runs out Sept. 30 would be no big deal, Mr. McConnell has been having none of it.

“Congress needs to address our nation’s most pressing needs with timely appropriations, and we need to keep the lights on come Oct. 1,” he said earlier this week.

Mr. McConnell sought to settle the scrutiny of his health this week by releasing a letter from the Capitol physician saying he had not suffered a stroke or seizure in his episodes. He vowed Wednesday to remain as leader at least through next year and serve out his Senate term ending in 2026.

Still, a shift in leadership probably would not mean a major immediate change in policy direction. His most likely successors share his general views on Ukraine and other spending matters and probably would not stray far. But they would have nowhere near his experience at navigating the politically treacherous currents ahead.

This is hardly a time, one lawmaker pointed out, for on-the-job training.