WASHINGTON — Senator Mitch McConnell is extending an open invitation to Senator Joe Manchin III — come on over to our side.

Mr. McConnell, the Kentucky Republican and minority leader, said on Tuesday that he was astonished by the angry response that Mr. Manchin of West Virginia elicited from the White House and his fellow Democrats with his Sunday bombshell that he would oppose President Biden’s signature domestic policy bill.

The Senate, Mr. McConnell noted, is an institution where the most important vote is the next one, leaving the Republican leader perplexed as to what drove Democrats to impugn Mr. Manchin’s integrity by accusing him of reneging on commitments to the president.

“Why in the world would they want to call him a liar and try to hotbox him and embarrass him?” Mr. McConnell, who is just one Senate seat away from regaining the majority leader title, asked in an interview. “I think the message is, ‘We don’t want you around.’ Obviously that is up to Joe Manchin, but he is clearly not welcome on that side of the aisle.”

It is hardly a secret, Mr. McConnell said, that he has wooed Mr. Manchin for years, only to have Mr. Manchin, a lifelong Democrat, resist. And Mr. Manchin this week said he “hoped” there was still a place for him in the party.

Despite his break with Mr. Biden over the sprawling safety net and climate change bill, Mr. Manchin would not be an exact fit for the Republican Party. He is closer to Republicans than Democrats on some flashpoint cultural issues like guns and abortion. But he has a more expansive view than most Republicans of the role of government in social and economic policy. And in both of former President Donald J. Trump’s impeachment trials, Mr. Manchin voted to convict.

But Mr. McConnell seemed to see the clash over the spending measure as potentially providing a new opening for a party switch that would both restore him as majority leader and shift the ground in Washington. And he is also not against stirring up trouble for Democrats however and whenever he can.

“Obviously we would love to have him on our team,” said Mr. McConnell. “I think he’d be more comfortable.”

Mr. McConnell’s appeal to Mr. Manchin came as the Republican leader celebrated the year coming to a close without Democrats advancing two of their most ambitious priorities: legislation to bolster voting rights and the sprawling domestic policy bill that Mr. McConnell characterized as part of a “socialist surge that has captured the other side.”

Considering how Republicans began 2021 — in the minority in Congress, a newly elected Democrat poised to move into the White House and a public worn down by a pandemic and alarmed by an assault on the Capitol — Mr. McConnell and his colleagues say they have had a successful year. In some respects, it was all the things they did not do that may have served them best.

They did not maneuver themselves into shutting down the government as they have in the past — despite demands from the right that they never work with Mr. Biden. And they did not allow the government to default, with Mr. McConnell providing Democrats a circuitous path to raising the debt ceiling. Either could have created a backlash for Republicans.

“We had a good deal of drama,” Mr. McConnell said about the high-wire act over the debt ceiling, “but in the end, we got the job done.”

As Democrats spent months trying to hammer out the huge policy bill among themselves, Republicans were relegated to the sidelines. Mr. McConnell said Democrats’ inability to come together on it so far reflected a misreading of the 2020 elections, when voters gave them the White House but bare majorities in both the Senate and the House.

“They did not have a mandate to do anything close to what they tried to do,” said Mr. McConnell, suggesting that progressive “ideology overcame their judgment.”

The decision by Mr. McConnell and other Republicans to help Democrats write and pass a separate, $1 trillion public works bill was, Mr. McConnell said, a smart one, even though Republican supporters of the measure took heat from others in the party, notably Mr. Trump.

Mr. McConnell said that applying pressure to keep the policy bill separate from the infrastructure measure denied Democratic leaders leverage over Mr. Manchin, who helped negotiate the infrastructure measure, while delivering a bipartisan bill that met legitimate needs. He said that Mr. Trump and other Republican critics had been proven wrong.

“I think it was a much smarter play to support the infrastructure bill,” he said. “I think it was, A, good for the country, and B, smart for us politically.”

Democrats have not given up on either the social policy bill or winning over Mr. Manchin, meaning Mr. McConnell and Senate Republicans will have to maintain their campaign against the legislation into the new year. Senator Chuck Schumer, the New York Democrat and majority leader, also intends to press forward with voting rights measures fiercely opposed by Mr. McConnell and is threatening to try to change Senate rules if Republicans try to filibuster it again.

Democrats say Mr. McConnell is being complicit in allowing some states to impose new voting restrictions meant to target voters of color, a charge he rejects, saying that the impact of the new laws is being exaggerated. He said he was relying on Senator Kyrsten Sinema, the Arizona Democrat who recently reaffirmed her opposition to changing filibuster rules, to hold steady.

“Kyrsten Sinema has been quite unequivocal that she is not going to break the Senate and eliminate the legislative filibuster,” he said. “Thank goodness for that.”

One area where Mr. Biden and Senate Democrats have posted some victories is on judicial confirmations, with 40 judges being seated in the president’s first year, a number well in excess of other recent presidents. It is a subject of special interest to Mr. McConnell, since he spent the Trump era pushing conservative judges through the Senate.

“Look, there’s some advantages to having the White House,” he said. “I’m not surprised, but the three Supreme Court justices and the 54 circuit judges we did are still there and I think will be good for the country for a long time to come.”

Mr. McConnell said he believed his party’s performance this year — and the struggles of the Democrats — were setting Republicans up for a strong midterm election next year and his potential return to running the Senate no matter what party Mr. Manchin is in. Despite Mr. Trump’s efforts to encourage candidates he favors in key Senate races, Mr. McConnell said he was intent on avoiding the type of primary contests that in the past have hurt Republicans by saddling them with primary winners who falter in general elections.

“I feel good about how we handled ourselves this year and I feel good about how the American public is reacting to what they are trying to do,” he said of the Democrats. “I believe it will be an excellent environment for us.”